Method for powering a magnetic coupler and device for powering an electric dipole

[0001] The present invention relates to a method and device for powering a multi-interphase transformer.

[0002] Multi-interphase transformers are used, for example, to connect a load to a multiphase power source. The following article provides further information on multi-interphase transformers: “Modelling and Analysis of Multi-Interphase Transformers for Connecting Power Converters in Parallel”, IN GYU PARK and SEON IK KIM, Dept. of Control and Instrumentation Eng., Wonkwang University, Iksan, Chonbuk, 570-749 Korea, IEEE 1997.

[0003] It is known to use multiphase power sources which are capable of generating N periodic supply voltages or currents which are offset angularly from one another, N being an integer greater than or equal to four. The angular offsets between the supply voltages or currents used are distributed uniformly between 0 and 2π rad. An angular offset of 2π rad corresponds to a period of the voltage or current.

[0004] Multi-interphase transformers known to the inventors can be broken down into at least four elementary magnetic cells, each cell comprising:

• • [0005] a magnetic core suitable for forming a single closed annular magnetic circuit, said core comprising for this purpose at least three non-co-linear bars forming the closed magnetic circuit, at least two of said bars having each a planar face facing the exterior of the cell, and the field lines of the closed magnetic circuit inside said bars being parallel to the planar faces,
  • [0006] one or more windings, each of said windings being wound around a bar of the magnetic core so as to leave at least the two bars with a planar face free of windings, and
  • [0007] the elementary cells are joined together in pairs via the respective planar faces thereof so as to form pairs of first and second cells which are magnetically coupled to one another.

[0008] The inventors are also familiar with multi-interphase transformers which can be broken down into at least four elementary magnetic cells, each cell comprising:

• • [0009] a magnetic core suitable for forming only a first and a second closed annular magnetic circuit with a common portion, said core comprising a central magnetic bar forming the common portion of the two closed magnetic circuits, and at least two non-co-linear bars each having a planar face facing towards the exterior of the cell, and the field lines of the first or second closed magnetic circuit inside said bars being parallel to the planar face thereof,
  • [0010] one or more windings, each of said windings being wound around the central bar so as to leave at least the two bars with a planar face free of windings, and
  • [0011] the elementary cells are joined together in pairs via the respective planar faces thereof so as to form pairs of first and second cells which are magnetically coupled to one another.

[0012] The methods of powering these multi-interphase transformers consist of:

• • [0013] a) powering the or each winding of the first cell with one of the supply voltages or currents so as to produce a magnetising flux in the bar of the first cell joined to the second cell, the fundamental component of which has an angular offset x_i, and
  • [0014] b) powering the or each winding of the second cell with one of the supply voltages or currents so as to produce a magnetising flux in the bar of the second cell joined to the first cell, the fundamental component of which has an angular offset x_j

[0015] The absolute value of the difference x_i-x_j is equal to

[0016] $\frac{2\pi }{N}.$

[0017] Multi-interphase transformers powered in this way function correctly but are bulky.

[0018] The object of the invention is therefore to propose a method of powering these multi-interphase transformers which allows the overall size of the multi-interphase transformers to be reduced while maintaining the same performance levels.

[0019] The invention therefore relates to a method of powering these multi-interphase transformers, in which the absolute value of the difference between the angular offsets x_i and x_j is greater than or equal to

[0020] $\frac{4\pi }{N}\mathrm{rad}.$

[0021] It has been found that applying an angular offset of greater than or equal to

[0022] $\frac{4\pi }{N}\mathrm{rad}$
between the angular offsets x_i and x_j reduces the maximum magnetic flux passing through the joined bars. Indeed, this brings the angular offset closer to π rad, which corresponds to an optimal reduction in the maximum magnetic flux which can be observed in the joined bars.

[0023] Since the maximum magnetic flux passing through the bars is reduced, it is also possible to reduce the dimensions of these bars in such a way that the overall size of the multi-interphase transformer is also reduced.

[0024] In addition, owing to the uniform distribution of the angular offsets of the N supply voltages or currents, the voltage or current harmonics in the load powered by this transformer are reduced.

[0025] The embodiments of this powering method may comprise one or more of the following features:

• • [0026] the absolute value of the difference between the angular offsets x_i and x_j is between

[0027] $\pi -\frac{2\pi }{N}\mathrm{rad}\mathrm{and}\pi +\frac{2\pi }{N}\mathrm{rad}$

• • [0028]  for each pair of cells;
  • [0029] each winding of a cell is connected in series with at least one other winding of another cell.

[0030] The invention also relates to a first embodiment of a device for powering an electric dipole, said device comprising:

• • [0031] a power source with N phases, the angular offsets between the phases being distributed uniformly between 0 and 2π rad, N being greater than or equal to four and 2π rad representing a period of the voltage or the periodic current,
  • [0032] a multi-interphase transformer which can be broken down into at least four elementary magnetic cells, each cell comprising:
  • [0033] a magnetic core suitable for forming a single closed annular magnetic circuit, said core comprising for this purpose at least three non-co-linear bars forming the closed magnetic circuit, at least two of said bars each having a planar face facing the exterior of the cell, and the field lines of the closed magnetic circuit inside said bars being parallel to the planar faces,
  • [0034] one or more windings, each of said windings being wound around a bar of the magnetic core so as to leave at least the two bars with a planar face free of windings, and
  • [0035] the elementary cells are joined together in pairs via the respective planar faces thereof so as to form pairs of first and second cells which are magnetically coupled to one another,
    in which:
  • [0036] a) the or each winding of the first cell is connected to a respective phase of the power source so as to produce, during operation, a magnetising flux in the bar of the first cell joined to the second cell, the fundamental component of which has an angular offset x_i, and
  • [0037] b) the or each winding of the second cell is connected to a respective phase of the power source so as to produce, during operation, a magnetising flux in the bar of the second cell joined to the first cell, the fundamental component of which has an angular offset x_j,
  • [0038] the absolute value of the difference between the angular offsets x_i and x_j is greater than

[0039] $\frac{4\pi }{N}\mathrm{rad}.$

[0040] The invention also relates to a second embodiment of a device for powering an electric dipole, said device comprising:

• • [0041] a power source with N phases, the angular offsets between the phases being distributed uniformly between 0 and 2π rad, N being greater than or equal to four and 2π rad representing a period of the voltage or the periodic current,
  • [0042] a multi-interphase transformer which can be broken down into at least four elementary magnetic cells, each cell comprising:
  • [0043] a magnetic core suitable for forming only a first and a second closed annular magnetic circuit with a common portion, said core comprising a central magnetic bar forming the common portion of the two closed magnetic circuits, and at least two non-colinear bars each having a planar face facing towards the exterior of the cell, and the field lines of the first or second closed magnetic circuit inside said bars being parallel to the planar face thereof,
  • [0044] one or more windings, each of said windings being wound around the central bar so as to leave at least the two bars with a planar face free of windings, and
  • [0045] the elementary cells are joined together in pairs via the respective planar faces thereof so as to form pairs of first and second cells which are magnetically coupled to one another,
    in which:
  • [0046] a) the or each winding of the first cell is connected to a respective phase of the power source so as to produce, during operation, a magnetising flux in the bar of the first cell joined to the second cell, the fundamental component of which has an angular offset x_i, and
  • [0047] b) the or each winding of the second cell is connected to a respective phase of the power source so as to produce, during operation, a magnetising flux in the bar of the second cell joined to the first cell, the fundamental component of which has an angular offset x_j,
  • [0048] the absolute value of the difference between the angular offsets x_i and x_j is greater than

[0049] $\frac{4\pi }{N}\mathrm{rad}.$

[0050] The embodiments of these power devices may comprise one or more of the following features:

• • [0051] the absolute value of the difference between the angular offsets x_i and x_j is between

[0052] $\pi -\frac{2\pi }{N}\mathrm{rad}\mathrm{and}\pi +\frac{2\pi }{N}\mathrm{rad}$

• • [0053]  for each cell;
  • [0054] each winding of the second cell is inferred from the corresponding winding of the first cells by means of axial symmetry along an axis which is co-linear with the joined faces;
  • [0055] each cell comprises at least one first and one second winding wound in opposite directions around the same bar;
  • [0056] each cell comprises at least one first and one second winding, the first winding and the second winding being connected to respective phases of the power source in such a way that, during operation, the angular phase difference between the supply voltages of each of said windings is between

[0057] $\pi -\frac{2\pi }{N}\mathrm{and}\pi +\frac{2\pi }{N}.$

[0073] FIG. 1 shows a device 2 for powering an electric dipole 4. In this case, the dipole 4 is connected to the device 2 by a filter 6 provided with an input 8.

[0074] The dipole 4 is a resistor for example.

[0075] The filter 6 is for example a filter comprising only a filter capacitor 12 connected in parallel with the terminals of the dipole 4. In this case, the device 2 enables a filter choke to be dispensed with.

[0076] The device 2 comprises a source 16 of multi-phase voltage and a multi-interphase transformer 18 to connect the source 16 to the dipole 4.

[0077] The source 16 is a source with N phases, N being an integer greater than or equal to 4. The source 16 thus supplies N voltages V_i, where the reference i is the number of the phase between 0 and N−1. By convention, the angular offset between the voltages V₀ and V_i is assumed to be

[0078] $\frac{2\pi i}{N}\mathrm{rad}.$
The angular onsets between the voltages V₀ to V_N−1 are thus uniformly distributed between 0 and 2π rad, as shown in FIG. 2.

[0079] In FIG. 2, each vector corresponds to a voltage Vi, the modulus of this vector corresponding to the modulus of the fundamental of the voltage and the angle of said vector to the x-axis corresponding to the angular offset thereof from the fundamental of the voltage V₀. As shown, when the angular offset of the fundamentals of the voltages V₀ to V_N−1 is uniformly distributed, the angular phase difference between two successive voltage vectors in the diagram in FIG. 2 is equal to 2π/N.

[0080] In this case, the amplitudes of the voltages V₀ to V_N−1 are all identical since all the voltages V₀ to V_N−1 have the same periodic waveforms which are offset from one another by an angular offset of

[0081] $\frac{2\pi }{N}\mathrm{rad}.$

[0082] In FIG. 1, the source 16 is shown in the form of N single-phase voltage sources S₀ to S_N−1 supplying the voltages V₀ to V_N−1. For example, the angular offset of the voltage generated by each source S_i can be adjusted so as to correspond to any one of the voltages V₀ to V_N−1. The voltages V₀ to V_N−1 are not generated in order by the sources S₀ to S_N−1, as described below.

[0083] For the purposes of simplification of FIG. 1, only three voltage sources S₀, S₁ and S_N−1 are shown.

[0084] The source 16 is for example a multiphase power supply network, a chopper or a multiphase voltage inverter, a controllable voltage rectifier formed from diodes and thyristors or a first stage of a flyback power supply. Said periodic voltages V_i are not necessarily sinusoidal but are rectangular or triangular, for example, and may comprise a continuous component.

[0085] In this embodiment, the multi-interphase transformer 18 comprises N single-phase transformers Tr₀ to Tr_N−1. Each transformer is formed by a primary winding e_1i and a secondary winding e_2i which are adjacent and magnetically coupled to one another via a magnetic core n_i, where i is the same reference as used above.

[0086] Each transformer forms a pair of windings which are magnetically coupled to one another by the magnetic core n_i.

[0087] In order to simplify the figure, only three transformers Tr₀, Tr₁ and Tr_N−1 are shown in FIG. 1.

[0088] One end of each primary winding e_1i is directly connected to the source S_i.

[0089] The secondary winding e_2i of each transformer Tr_i is connected to the source S_i−1 by the primary winding e_1,i−1 of the transformer Tr_i−1. When i is equal to zero, the secondary winding e₂₀ is connected to the source S_N−1 by the winding e_1,N−1 of the transformer Tr_N−1.

[0090] The end of each secondary winding not connected to one of the sources S_i is directly connected to a common point 24 which is itself directly connected to the input 8 of the filter 6.

[0091] The multi-interphase transformer 18 will now be described in greater detail with reference to FIGS. 3 and 4 for the specific case in which the number N of phases is twelve.

[0092] FIG. 3 is a cross-section of the multi-interphase transformer 18. This multi-interphase transformer 18 is formed from twelve elementary magnetic cells C₀ to C₁₁ joined to one another in a horizontal direction L. Each cell C_i corresponds to a single-phase transformer Tr_i.

[0093] Two adjacent cells C_i and C_j are shown in greater detail in FIG. 4.

[0094] Each cell C_i comprises a magnetic core n_i with a cross-section in the shape of a ladder or an “8”. To this end, the magnetic core is formed from six lateral bars B_1,i to B_6,i and a central bar B_c,i. The bars B_1,i and B_2,i form the left side-leg M_Gi of the ladder. The bars B_4,i and B_5,i form the right side-leg M_Di. The side-legs M_Gi and M_Di may be formed from a single piece.

[0095] The bar B_ci is a central horizontal bar whereas the bars B_3,i and B_6,i are horizontal bars located at the top and bottom respectively of the side-legs M_Gi and M_Di.

[0096] The cross-section of each of the side-legs or bars is substantially rectangular.

[0097] More specifically, the lateral bars B_1,i to B_6,i each have a planar face, F_1,i to F_6,i respectively, which face towards the exterior of the cell C_i.

[0098] The core n_i has two windows or apertures 32 and 34 which are located above and below the central bar B_Ci respectively.

[0099] The cell C_i also comprises two windings 36 and 38 which are wound around the central bar B_Ci. The windings 36 and 38 are wound in opposite directions. Each winding preferably comprises a plurality of turns.

[0100] The winding direction of the turns of each winding is defined by a dot surrounded by a circle and a circle containing a cross. The dot surrounded by a circle indicates that a vector exits the plane of the page, whereas a circle containing a cross indicates that this vector enters into the plane of the page.

[0101] In the following description it shall be considered that the windings wound in a clockwise direction as viewed from the right-hand side of the multi-interphase transformer 18 in FIG. 3 rotate in a positive direction. The windings wound in the opposite direction rotate in a negative direction. The two following references “V_i” and “−V_i” are defined accordingly. “V_i” is the supply voltage of a winding wound in the positive direction “−V_i” is the supply voltage of a winding wound in the negative direction.

[0102] Each of said windings 36, 38 corresponds to a winding e_2i or e_1i of the multi-interphase transformer 18. For this reason, each winding of a cell bears the reference e_1i or e_2i in FIG. 3. Furthermore, only the direction of winding of each winding has been shown in FIG. 3.

[0103] The core n_i concentrates the field lines of the magnetic field created by the windings 36 and 38. These field lines form a magnetising flux. In FIG. 4, two arrows represent two field lines of the magnetising flux E_Hi and E_Bi created by the windings 36 and 38 within the core n_i. These arrows also represent the following sign convention: when the amplitude of the fundamental of the magnetising flux E_Hi is positive, the lines of this field E_Hi are considered to rotate in the positive direction if they rotate in a clockwise direction. When the amplitude of the fundamental of the magnetising flux E_Bi is positive, the lines of this field E_Bi are considered to rotate in the positive direction if they rotate in an anticlockwise direction. This sign convention applies to all of the cells C_i of the multi-interphase transformer. The reference w_i is also used to denote the angular offsets of the fundamental components of the magnetic fluxes E_Hi and E_Bi from the same reference. Using this sign convention, it is possible to indicate that the same magnetising flux can be defined as moving in the positive direction with an offset of w_i or as moving in the negative direction with an offset w_i+π.

[0104] More specifically, the field line of the flux E_Hi enters from the right of the central bar B_Ci and is closed by passing through the bar B_6i at the top when it rotates in a positive direction. The field line E_Bi also enters from the right of the bar B_6i and is closed by means of the bar B_3i at the bottom when said line rotates in the positive direction. These field lines E_Hi and E_Bi correspond to a magnetising flux created by the windings 36 and 38.

[0105] The core n_i thus enables two closed magnetic circuits to be formed. These closed magnetic circuits have a common portion, i.e. the bar B_ci.

[0106] As will be explained below, the amplitude and the phase of the fundamental component of this magnetising flux is a function of the angular offsets of the supply voltages of the windings 36 and 38.

[0107] The cell C_j can be inferred from the cell C_i on account of axial symmetry. The cell C_j is thus constructionally identical to the cell C_i with the exception that the positions of the windings 36 and 38 have been swapped in relation to the positions of the windings 36 and 38 of cell C_i.

[0108] When the windings of the cell C_j are supplied with power, field lines E_Hi+1 and E_Bi+1 are formed in the core 30 of the cell C_j. The same sign convention as defined for cell C_i also applies to cell C_j.

[0109] The entire magnetic field generated by the windings 36 and 38 is not concentrated within the core n_i. As shown by the arrows F_Hi and F_Bi, magnetic field leakage lines are formed around the winding 36. These lines correspond to a magnetic leakage flux. Unlike the magnetising flux, the magnetic flux leakage lines comprise at least one portion which extends outside the core n_i. For example, in this case, the magnetic flux leakage lines pass through the windows 32, 34 in order to close. The windows 32, 34 are formed from air for example.

[0110] In this embodiment, the faces F_4,i, F_5,i, of the cell C_i are joined, and more specifically are respectively brought into contact with the faces F_2,j and F_1,j of the cell C_j. The magnetising fluxes E_Hi, E_Hj and E_Bi, E_Bj thus merge in the side-legs M_Di and M_Gj.

[0111] The side-legs M_Di and M_Gj are for example adhesively bonded or connected to one another by any means enabling close contact to be maintained between the two side-legs.

[0112] The arrows S_H and S_B define a sign convention which applies to all the magnetising fluxes circulating in the joined bars. More specifically, this common sign convention enables the angular offsets of the magnetising fluxes in each cell to be compared.

[0113] In this sign convention:

• • [0114] x^d_i indicates the angular offset of the magnetising fluxes E_Hi and E_Bi in the bar of the cell C_i joined to the cell C_j, and
  • [0115] x^g_j indicates the angular offset of the magnetising fluxes E_Hj and E_Bj in the bars joined to those of the cell C_i. These references make it possible to note that, in FIG. 4, the offset x^d_i is equal to the offset w_i since the sign conventions used to define these offsets w_i and x_i are in the same direction. Conversely, the offset x^g_j is equal to the offset w_j+π rad since the sign conventions used to define the offsets w_j and x^g_j are in the opposite direction.

[0116] FIG. 4 likewise shows a face O_i,j located at the intersection of the uprights M_Di and M_Gj and perpendicular to the plane of the page. There are magnetising fluxes E_Bi and E_Bj across this face O_i,j. With an identical direction of displacement for the fluxes E_Bi and E_Bj, the greater the difference between the angular offset x_i of the fundamental of the magnetising flux E_Bi and the angular offset x_j of the fundamental of the magnetising flux E_Bj is, the lower the maximum magnetising flux across the face O_i,j is. Also, the lower the magnetising flux across the face O_i,j is, the more the horizontal section of the bars B_4,i and B_2,j can be reduced, and this reduces the overall size of the multi-interphase transformer 18. The same explanations apply to the reduction of the overall size of the bars B_5,i and B_1,j.

[0117] The multi-interphase transformer 18 only comprises cells coupled in pairs in the horizontal direction L, i.e. coupled to one another via the faces of the uprights thereof, as was described with reference to FIG. 4.

[0118] The design and functionality of the device 2 will now be described with reference to FIG. 5.

[0119] Initially, in a step 40 of designing the multi-interphase transformer 18, elementary magnetic cells which are all identical to one another are produced. For example, at this stage each of the cells is identical to the cell C_i described with reference to FIG. 4. Subsequently, the following rules are applied:

• [0120] a) the supply voltages V_i of each winding of a cell C_i are selected such that an angular offset α_i between the supply voltages of the two windings of said cell is between

[0121] $\pi -\frac{2\pi }{N}\mathrm{and}\pi +\frac{2\pi }{N}\mathrm{rad},$

• [0122]  and
• [0123] b) each cell C_i is coupled to the adjacent cell C_j, which, before gluing, generates in the bars of the left upright B_1j and B_2j the magnetic fluxes E_Hj and E_Bj, of which the angular offset x_j is between

[0124] $x_i+\frac{2\pi }{N}+\pi \mathrm{and}{x}_i-\frac{2\pi }{N}+\pi .$

[0125] Rule a) corresponds to the application of the teaching of the French patent application filed as FR 05 07 136 and makes it possible to limit the overall size of the multi-interphase transformer 18 even further.

[0126] By way of example, the application of rule a) leads to the selection of the voltage pair V_i shown in the following Table for each cell:

Table 1: see annex at the end of the description.

[0127] In Table 1, the symbol C_i indicates the cell. In each column C_i, the symbols V_i on the left and right identify the corresponding voltages of the windings, on the left and right respectively, of the cell C_i.

[0128] Since the supply voltages V_i all have the same amplitude, and the angular offset α_i being the same for each of the cells C_i, the amplitude of the fundamental of the magnetising flux generated by each of the cells is thus identical. Thus, the application of rule b) consists of:

• [0129] 1) estimating, in an operation 42, the angular offset w_i of the magnetising flux generated by each cell C_i from the supply voltages in Table 1, and
• [0130] 2) joining, in an operation 44, each cell C_i to another cell C_j in such a way that the absolute value of the difference γ between the angular offsets x_i and x_j of the magnetising fluxes in the joined bars is between

[0131] $\pi -\frac{2\pi }{N}\mathrm{and}\pi +\frac{2\pi }{N}.$

[0132] The estimation of the angular offset w_i will now be explained using FIG. 6 in the particular case of the cell C₀.

[0133] FIG. 6 corresponds to the graph in FIG. 2 in the case where N is equal to twelve.

[0134] Based on Table 1, the voltages V₀ and V₅ are used to power the windings 36 and 38 respectively of the cell C₀.

[0135] The angular offset w_i of the magnetising flux can be estimated as the vector sum of the voltage vector {right arrow over (V)}₀ and the vector −{right arrow over (V)}₅. The result of this vector summation is shown in FIG. 6 by a dashed arrow F. This arrow F makes an angle w of

[0136] $\frac{-\pi }{12}\mathrm{rad}$
with the x-axis. This angle w corresponds to an estimate of the angular offset w₀.

[0137] It will be noted that to obtain this result it is necessary to take the inverse of the voltage vector {right arrow over (V)}₅, because the winding 38 is wound in the negative direction.

[0138] In the case of the multi-interphase transformer 18, at the end of the operation 42, Table 2 is obtained, the second line of which shows the offsets w_i:

[0139]

TABLE 2
C0	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11
- π 12	9 ⁢ π 12	19 ⁢ π 12	5 ⁢ π 12	15 ⁢ π 12	π 12	11 ⁢ π 12	21 ⁢ π 12	7 ⁢ π 12	17 ⁢ π 12	3 ⁢ π 12	13 ⁢ π 12

[0140] Subsequently, during the operation 44, the cells C_i are classified by increasing or decreasing order of angular offset w_i.

[0141] In the present case, the list of cells classified by increasing order of angular offset w_i is as follows:

• [0142] {C₀, C₅, C₁₀, C₃, C₈, C₁, C₆, C₁₁, C₄, C₉, C₂, C₇}.

[0143] At this stage, in a first embodiment, the cells are joined to one another in the horizontal direction L in the increasing order shown above. Thus, the bars B_5i and B_4i of the cell C_i are joined to the bars B_1,j and B_2,j respectively of the following cell C_j.

[0144] In this first embodiment, it is checked that the difference γ between the angular offsets x^d_i and x^g_j is approximately π rad. For example, the angular offset w₀ is in this case equal to −π/12 whilst the angular offset w₅ is equal to π/12 rad. The offset x^d_i in the bar B₅₀ is therefore equal to −π/12 rad. The offset x^g₅ in the bar B₁₅ is equal to w₅+π, i.e. π/12+π, because in the bar B₁₅, the sign convention adopted for defining the offset of the flux E_H5 is in the opposite direction of the arrow S_H. The difference γ is therefore equal to π+2π/12 rad in this case.

[0145] In the first embodiment, the offsets w_i are distributed over 360° and the continuous components of the magnetic fluxes in the joined bars do not cancel out.

[0146] To obtain a second embodiment of the multi-interphase transformer, in which the differences γ are even closer to π rad, a step 46 of permuting the supply voltages is performed.

[0147] More precisely, in step 46, using Table 2, the cells are divided into two halves in such a way that the angular offset w_i of each cell in the first half is less than all the angular offsets w_i of the cells in the second half. For example, in this case, the second half consists of the last six cells in the classification in increasing order shown above, i.e. in this case the cells C₆, C₁₁, C₄, C₉, C₂ and C₇.

[0148] Subsequently, the supply voltage of the left winding of each of the cells in this second half is permuted with the supply voltage of the right winding of the same cell. This permutation of the supply voltages does not alter the position of the windings 36 and 38. Thus, using the notations defined with reference to Table 1, this operation of permuting the voltages of a cell makes it possible to pass from the couple of supply voltages (V_a, −V_b) to the supply couple (V_b, −V_a).

[0149] In contrast, the operation of permuting the voltages reverses the direction in which the fundamental component of the magnetic flux rotates. The angular offset w_i of each permuted cell is thus incremented by π rad. The second line of Table 3 below shows the angular offset w_i of each cell C_i at the end of step 46:

[0150]

TABLE 3
C0	C5	C10	C3	C8	C1	C6	C11	C4	C9	C2	C7
- π 12	π 12	3 ⁢ π 12	5 ⁢ π 12	7 ⁢ π 12	9 ⁢ π 12	- π 12	π 12	3 ⁢ π 12	5 ⁢ π 12	7 ⁢ π 12	9 ⁢ π 12

[0151] It can thus be seen that the angular offsets of the set of cells are now distributed over 180°, and no longer over 360°.

[0152] Subsequently, still in step 46, the cells C_i thus obtained are again classified by increasing order of angular offset w_i. The following classification is thus obtained:

• [0153] {C₀, C₆, C₅, C₁₁, C₁₀, C₄, C₃, C₉, C₈, C₂, C₁, C₇}.

[0154] In this second embodiment, the cells C_i thus obtained are then joined in the horizontal direction L in the order shown above.

[0155] The first line of Table 4 (see annex) shows the order of the cells in this second embodiment. The second line shows the voltage at which each of the windings of each cell is connected.

[0156] It is checked that in the second embodiment the difference γ obtained is closer to π rad than in the first embodiment. For example, the angular offsets w₀ and w₆ of the joined cells C₀ and C₆ are both equal to −π/12 rad. Consequently, the angular offset x^d₀ in the bar B₅₀ before gluing is equal to −π/12 rad. The angular offset x^g₆ in the bar B₁₆ before gluing is equal to −π/12+π. In fact, as before, the sign convention for defining the offset w₆ is in the opposite direction from the arrow S_H. Thus, in the bar B₁₆, the offset x^g₆ is given by the following relationship: x^g₆=w₆+π.

[0157] The difference γ is equal to π rad.

[0158] It will also be noted that the difference γ between the joined bars of the cells C₆ and C₅ is equal to π+2π/12 rad.

[0159] Thus, the maximum amplitude of the fundamental of the magnetising flux in the joined uprights of a pair of cells is substantially zero. Moreover, the maximum amplitude of the magnetising flux generated within the joined uprights of two cells belonging to different pairs is greatly reduced. This makes it possible to reduce greatly the section of these uprights and therefore the overall size of the multi-interphase transformer 18.

[0160] This second embodiment therefore makes it possible to make the difference γ even closer to π rad. However, neither the first nor the second embodiment makes it possible to cancel the continuous components of the magnetising fluxes in the joined bars.

[0161] To cancel this continuous component in the joined bars, a step 48 of permuting the positions of the coils is then performed. The step 48 is in this case applied to half of the cells. Step 48 may be performed after step 46 or straight after step 40.

[0162] In step 48, the first two cells of the list of cells classified by increasing order are grouped to form a first pair, then the following two cells are grouped to form a second pair, and so on.

[0163] Subsequently, for the second element of each pair, the position of the two coils of the cell is permuted relative to the position on which the classifying operation was based. For example, in the case of the cell C₆, in the operation 46, the windings 36 and 38 are on the left and right of the cell respectively. Once the positions have been permuted, the windings 36 and 38 are on the right and left of the cell respectively.

[0164] Subsequent to a permutation of this type, the windings immediately to the right and left of a joined bar are wound in the same direction. A configuration of this type reduces or cancels the continuous components of the magnetising fluxes in the joined bars. Moreover, the permutation of the position of the windings does not alter the angular offset w_i of the cell, because the windings 36 and 38 are still supplied with the same voltages. Finally, only the position of the windings is altered. Under these conditions, step 48 makes it possible, in addition to the reduction of the maximum amplitude of the fundamental component of the magnetising flux in the joined bars, also to reduce the continuous component in said joined bars.

[0165] For example, Table 5 below shows the supply voltages, of each cell of the multi-interphase transformer, obtained at the end of steps 46 and 48. The first line shows the order of the cells. The second line shows the supply voltages of the left and right coils of each cell.

Table 5 (See Annex)

[0166] After design, in a step 50, the angular offset of each supply S₀ to S_N−1 is controlled in such a way that the supply voltage of the first windings e_1i, e_2i of each cell C_i corresponds to that which was determined in the design step 40.

[0167] Consequently, in a step 52, the windings of each cell are powered using the source 16 controlled in this way. This makes it possible to supply the dipole 4 from a multiphase supply.

[0168] The multi-interphase transformer 18 has been described in the particular case where it is formed of twelve cells. However, what has been described above is applicable to any multi-interphase transformer formed of at least four cells. By way of example, Tables 6 and 7 below describe the configuration of multi-interphase transformers having from 4 to 20 cells obtained by performing operations 46 and 48.

[0169] More precisely, in each of these Tables 6 and 7, column “N” shows the total number of cells and the following columns show what supply voltages are to be used for each cell C_i to power the windings thereof. In these Tables, each column C_i is divided into two sub-columns. The left and right sub-columns show the supply voltage at which the left and right windings respectively of this cell must be connected. In these sub-columns, the absolute value of the number j shown indicates that this winding must be supplied with the voltage V_j−1. The “−” symbol before the number j simply indicates that this winding is wound in the negative direction.

Table 6 (See Annex)

Table 7 (See Annex)

[0170] FIG. 7 shows a multi-interphase transformer 50 which can be used instead of and in the place of the multi-interphase transformer 18 in the device 2.

[0171] This multi-interphase transformer 50 likewise comprises twelve cells C_i which are respectively identical to the cells C_i of the multi-interphase transformer 18. In contrast to the multi-interphase transformer 18, the cells C_i of the multi-interphase transformer 50 are joined not only in the horizontal direction L, as in the multi-interphase transformer 18, but also in a vertical direction H.

[0172] More precisely, each pair of cells (C₀; C₆); (C₅; C₁₁); (C₁₀; C₄); (C₃; C₉); (C₈; C₂) and (C₁; C₇) is joined in a horizontal direction via the respective vertical uprights thereof. Moreover, each pair of cells is also joined in the vertical direction H to another pair of cells via the horizontal bars thereof. The cells are joined in the vertical direction in the same way as in the horizontal direction, i.e., for example, by directly contacting the planar faces of the lateral bars of these cells. These cells may be fixed in the vertical direction by gluing or by any other means.

[0173] As in the case of the multi-interphase transformer 18, the supply voltages of each of the windings of each cell are determined as a function of the angular offsets w_i of the magnetising fluxes.

[0174] For example, when the multi-interphase transformer 50 is designed, the procedure is the same as was described with reference to operations 42 to 48 until Table 5 is obtained. Then, alternately, for every other cell, the following operations are carried out:

• • [0175] permuting the supply voltages of each of the cells of these pairs, and
  • [0176] permuting the positions of the windings of each of these cells.

[0177] In this case, these operations are therefore applied to the following pairs of cells: {C₅; C₁₁}, {C₃; C₉} and {C₁; C₇}.

[0178] The permutation of the supply voltages is identical to operation 46 and thus makes it possible to add an offset of π rad, allowing the fundamental component to be cancelled in the joined horizontal bars.

[0179] The step of permuting the windings is identical to operation 48 and thus makes it possible to cancel the continuous component of the magnetising flux in the joined horizontal bars.

[0180] At the end of these steps, the distribution of voltages shown in the Table below is obtained for each cell:

Table 8 (See Annex)

[0181] It is then checked that the difference γ between, for example, the offset x₅ in the bar B₆₅ and the offset x₀ in the bar B₃₀ is approximately π rad. In this case, w₀ and w₅ are equal to −π/12 and π/12+π respectively. If the common sign convention for the bars B₃₀ and B₆₅ is such that the magnetising flux is positive when it is displaced from left to right in the joined bars, then x^b₀=−π/12 rad and x^h₅=π/12+π, where x^b₀ and x^h₅ are the angular offsets in the bars B₃₀ and B₆₅ respectively. Therefore the difference γ is equal to π rad.

[0182] Thus, in the multi-interphase transformer 50, the overall size of the joined vertical uprights and of the joined horizontal bars can be greatly reduced.

[0183] FIG. 8 shows another embodiment of the cells C′_i and C′_j which can be used instead of and in the place of the cells C_i and C_j respectively.

[0184] The construction of the cell C′_i is identical to that of the cell C_i except that the winding 36 is wound around the winding 38 and not to the side. For example, the winding 36 is wound on the periphery of the winding 38.

[0185] For example, the cell C′_j is identical to the cell C′_i.

[0186] FIGS. 9 to 11 show cells A_i, A′_i and A″_i respectively, comprising a core 60 having an annular cross-section in a vertical plane. In this case, the core 60 is formed of two horizontal bars and two vertical bars.

[0187] These cells A_i, A′_i and A″_i each comprise only two windings 62 and 64 wound in the opposite direction from one another. In the cells A_i and A″_i, the windings 62 and 64 are wound around the same bar. In the cell A_i, the winding 62 is only wound around an upper part of the vertical bar, whilst the winding 64 is only wound around a lower part of the same bar.

[0188] In the cell A″_i, the winding 64 is wound around the winding 62 and preferably around the periphery of the winding 62.

[0189] In the cell A′_i, the winding 62 is only wound around a vertical bar of the cell, whereas the winding 64 is only wound around the other vertical bar of the cell.

[0190] The free bars of each coil of the cells A_i, A′_i and A″_i each have a planar face pointing towards the outside of the cell. These planar faces make it possible to join the cells to one another to form a multi-interphase transformer. The supply voltage of each of these windings is selected as a function of the angular offset x_i of the fundamental of the magnetising flux concentrated in the joined bars. For this purpose, the teaching provided with reference to FIG. 5 is adapted for the case involving the cells A_i, A′_i and A″_i so as to minimise the maximum amplitude of the fundamental of the magnetising flux in these bars which are joined to one another.

[0191] Finally, the cells A_i, A′_i and A″_i are basically distinguished from the cells C_i and C′_i in that in the cells A_i, A′_i and A″_i, a single annular closed magnetic circuit is established, whereas in the cells C_i and C′_i, two annular closed magnetic circuits are established on different paths.

[0192] FIG. 12 shows another supply device 70 for the electric dipole 4. For this purpose, this device 70 comprises the power source 16 as well as a multi-interphase transformer 72 for connecting the N phases of the source 16 to the dipole 4.

[0193] More precisely, the multi-interphase transformer 72 comprises N windings L_i producing inductance. Only one side of each winding L_i is connected to the source S_i, the other side being connected to the common point 24.

[0194] The construction of the multi-interphase transformer 72 is described in greater detail with reference to FIG. 13 in the particular case where the number N of phases is equal to five.

[0195] The multi-interphase transformer 72 is produced by joining five identical elementary magnetic cells B₀ to B₄ in the vertical direction H. The cell B_i is described in greater detail with reference to FIG. 14.

[0196] In this example, the cell B_i comprises a magnetic core 74 having an anular cross-section. This core 74 is formed of only two vertical bars and two horizontal bars. In this case, the three bars without coils each have a planar face pointing towards the outside and allowing this cell to be magnetically coupled to another cell. At least one of the bars comprises a gap 75 for preventing saturation of the core 74 caused by a continuous component of the magnetising flux.

[0197] The cell B_i likewise comprises only a single winding 76, wound around only one of the vertical bars. This winding 76 generates a magnetising flux E_i concentrated in the interior of the core 74. A single field line of the magnetising flux E_i is shown in FIG. 14. This magnetising flux has an angular offset w_i which is a function of the angular offset of the supply voltage of the winding 76. More precisely, in the case of the cell B_i, the estimate of the angular offset w_i is set equal to the angular offset of the supply voltage of the winding 76.

[0198] In FIG. 13, the cells B_i are joined to one another so as to be juxtaposed edge to edge with the planar faces of the respective horizontal bars thereof.

[0199] As before, the angular offset of the supply voltage of the windings of the cell B_i is determined in such a way as to minimise the amplitude of the fundamental of the magnetising flux circulating in the joined bars. More precisely, the supply voltages of joined cells B_i and B_j are selected in such a way that the difference between the angular offsets x_i and x_j of the fundamentals of the magnetising fluxes generated by each of these cells is as close as possible to π rad.

[0200] The method described with reference to the procedure of FIG. 5 is adapted to produce the multi-interphase transformer 72. For example, in FIG. 13, the windings of the cells B₁ to B₄ are all wound in the same direction. Thus, in this configuration, the windings of the cells B₀ to B₄ are supplied with the voltages V₁, V₃, V₅, V₂, and V₄ respectively.

[0201] FIG. 15 shows the architecture of a cell D_i having a core identical to the core n_i of the cell C_i. The cell D_i is only provided with a single winding 80 wound around the central bar. The central bar comprises a gap 81 for avoiding the saturation of the core n_i caused by a continuous component of the magnetising flux. This cell D_i can be used instead of and in the place of the cell B_i in multi-interface transformers similar to the multi-interphase transformer 72.

[0202] FIG. 16 shows a third embodiment of a device 90 for supplying the dipole 4. In this figure, the elements already described with reference to FIG. 1 have the same reference numerals, and only the differences from the device 2 are discussed here.

[0203] In FIG. 16, the filter 6 does not have to exhibit any inductance.

[0204] The device 90 comprises the power source 16 connected to the dipole 4 via a multi-interface transformer 92.

[0205] In the multi-interphase transformer 92, the central point 24 is connected to a reference potential M₁ and no longer to the input 8 of the filter 6.

[0206] In this embodiment, each transformer Tr_i comprises, in addition to the pair of windings e_1i and e_2i, a pair of windings e_3i and e_4i. The windings e_3i and e_4i are magnetically coupled to the windings e_1i and e_2i via the magnetic core n_i. The pair of windings e_3i and e_4i is electrically insulated from the windings e_1i and e_2i.

[0207] One end of the winding e_3i is connected via a diode d_i to a common point 96. The cathode of the diode d_i is directed towards the common point 96.

[0208] The common point 96 is directly connected to the input 8 of the filter 6.

[0209] The other end of the winding e_3i is directly connected to one end of the winding e_4,i+1 of the following transformer Tr_i+1. The end of the winding e_4,i+1 which is not connected to the winding e_3i is connected to a reference potential M₂ electrically insulated from the potential M₁.

[0210] The end of the winding e_3,N−1 which is not connected to the common point 96 is directly connected to one end of the winding e₄₀.

[0211] The design and supply procedure for the multi-interphase transformer 92 is the same as that described with reference to FIG. 5, so as to reduce the overall size of said multi-interphase transformer. In particular, the estimate of the angular offset w_i of a cell is obtained, using only the supply voltages of the windings e_1i and e_2i. In order to return to the previous case of two windings per cell, it is possible to equate the parts of the windings e_1i, e_3i and e_2i, e_4i respectively to the windings e_1i and e_2i of FIG. 1.

[0212] FIG. 17 shows an example of cells E_i which can be used to form the multi-interphase transformer 92. This cell E_i is identical to the cell A′_i except that the windings 62 and 64 have been doubled up. In FIG. 17, the doubles of the windings 62 and 64 have the reference numerals 102 and 104 respectively.

[0213] The windings 102 and 104 are wound around the core 60 in the same direction as the windings 62 and 64 respectively. These windings 102 and 104 are electrically insulated from the windings 62 and 64 and magnetically coupled to these windings via the core 60. The windings 62 and 64 correspond to the windings e_1i and e_2i respectively of FIG. 16 and the windings 102 and 104 correspond to the windings e_3i and e_4i respectively of FIG. 16.

[0214] FIG. 18 shows the construction of a cell F_i which can likewise be used to produce the multi-interphase transformer 92.

[0215] This cell F_i is identical to the cell C_i except that the windings 36 and 38 have been doubled up. The doubles of the windings 36 and 38 have the reference numerals 106 and 108 respectively. In this embodiment, the winding 106 is only wound around the winding 36 and the winding 108 is only wound around the winding 38. The windings 106 and 108 are electrically insulated from the windings 36 and 38 and magnetically coupled to these windings 36 and 38 via the core n_i.

[0216] FIG. 19 shows the architecture of a DC-DC converter using a multi-interphase transformer as described with reference to the previous figures.

[0217] The converter 110 comprises a continuous power source 122 connected to the input of an inverter 124 which converts the continuous voltage provided by the source 122 into N periodic tensions which are angularly offset from one another by

[0218] $\frac{2\pi }{N}\mathrm{rad}.$

[0219] The inverter 124 is in this case a bidirectional current inverter. This inverter is known and the construction thereof will not be described in detail here.

[0220] The connection of the source 122 and the inverter 124 thus forms a multiphase power source 126. The source 126 is connected to a multi-interphase transformer 128 with galvanic insulation similar to that of the multi-interphase transformer 92. However, in this embodiment, one end of each winding e_1i and e_2i is connected directly to a respective phase of the source 126. The other ends of each of these windings e_1i and e_2i are electrically interconnected.

[0221] One end of each of the windings e_3i and e_4i is connected to a respective input of a voltage rectifier 130. The other end of these windings e_3i and e_4i is connected to a reference potential M₃.

[0222] The rectifier 130 comprises the same number of branches and of inputs receiving the voltage supplied by the winding e_3i and e_4i. Each branch is formed of a controlable switch I_i and a diode D_i connected in parallel. The controlable switch I_i is a switch which only allows current to circulate in one direction from the input which is connected to the winding e_3i or e_4i to a common point 134. The different controlable switches of the rectifier 130 are controlled in such a way as to rectify the voltage supplied by each of the windings e_3i and e_4i. In this embodiment, the dipole 4 is connected between the common point 134 and the reference potential M₃.

[0223] The rectifier 130 is in this case a bidirectional current rectifier.

[0224] FIG. 20 shows another embodiment of a DC-DC converter 140. This converter 140 comprises a multiphase power source 142 made up of a continuous voltage source 144 connected to the input of an inverter 146. In this case, for example, the inverter 146 has unidirectional current. Each output of the inverter 146 is connected to a respective winding of a multi-interphase transformer 148. The multi-interphase transformer 148 is identical to the multi-interphase transformer 128 except that in the multi-interphase transformer 128, it is the ends of the windings e_3i and e_4i that are connected to the respective phases of the source 142. For this reason, during the operation of estimating the angular offset x_i of each of the cells, the windings e_3i and e_4i as well as the supply voltages thereof are to be taken into account.

[0225] In this case, the outputs of the multi-interphase transformer 148 are connected to a voltage rectifier/step-up transformer 149. For example, the voltage rectifier/step-up transformer 149 may be formed of a plurality of step-up transformer stages 150 to 153. Each step-up transformer stage receives the voltages generated by a pair of coils e_1i, and e_2i respectively in order to step up a volage received at the input. A rectifier/step-up transformer of this type is known and therefore will not be described in greater detail. The load 4 is connected to the output of this rectifier/step-up transformer 149.

[0226] Many other embodiments are possible. Here, each multi-interphase transformer has been described in the case where it is produced by gluing or fixing a plurality of elementary magnetic cells to one another. In a variant, the multi-interphase transformer has exactly the same construction as that described here, but is produced by joining a succession of magnetic cores behind one another in an E-shape. More precisely, the free ends of the horizontal bars of the E-shaped portion are joined to the vertical rear face of the following E-shaped core. The free ends of the horizontal bars of the last E-shaped core in the stack are magnetically connected via a vertical I-shaped bar. The construction of the multi-interphase transformer thus obtained is identical to that obtained by joining cells such as the cells C_i, for example. Thus, a multi-interphase transformer of this type can be broken down into elementary cells identical to those described here. From this point, it is possible to locate, within this multi-interphase transformer, portions of the core corresponding to each of the bars B_ij. However, in this case, the joined bars B_ij and B_ij+1 come from the same material as one another, i.e. are formed in a single block. The teaching described previously can therefore be applied to a multi-interphase transformer of this type to determine what phase of the power source each winding must be connected to in order to minimise the maximum magnetic flux in the joined bars.

[0227] It is also possible to produce a multi-interphase transformer, having one of the constructions described here, from a core in a single block in which the number of openings provided is the same as the number of windows 32, 34 required. In this last embodiment, no gluing between different cells is required. However, a multi-interphase transformer of this type can nevertheless still be broken down into elementary cells of the type described here. Thus, the teaching provided in this description can be applied to this embodiment to determine what phases of the voltage source each of the windings must be connected to in order to minimise the maximum magnetic flux in the joined bars.

[0228] A multi-interphase transformer with reduced overall size may also be produced by joining a plurality of cells in the vertical direction alone.

[0229] In this case, the multi-interphase transformer comprises the same number of windings e_1i and e_2i and phases of the power source. In a variant, each winding e_1i and e_2i is divided into a plurality of windings e_1ik and e_2ik respectively which are connected in series. Each winding e_1ik and e_2ik is subsequently used in a different cell. However, the number of windings e_1ik and e_2ik connected in series preferably remains less than N.

[0230] The embodiments of the multi-interphase transformers described here are based on the particular case where the teaching of the patent application FR 05 07 136 is used within each cell in such a way as to reduce even further the overall size of each of these cells (rule a) as described with reference to FIG. 5). However, in a variant, only rule b) as described with reference to the same figure is used to reduce the overall size of the multi-interphase transformer.

[0231] If a cell comprises two windings, the turns of these windings may be interlaced. This decreases the alternative resistance of the windings.

[0232] The planar face of the joined bars may comprise roughnesses or irregularites making it easier to couple and fix the cells to one another.

[0233] The bars around which coils are wound are not necessarily straight but may be curved.

[0234] Finally, it is also possible to use different types of cells in a single multi-interphase transformer.

[0235] The different embodiments described here have the following advantages:

• • [0236] the application of rule b) for each joined bar makes it possible to reduce very distinctly the overall size and losses of the multi-interphase transformer,
  • [0237] selecting the supply voltages of the windings in such a way that the difference between the angular offsets x_i and x_j of the magnetising fluxes generated by two joined cells is between

[0238] $\pi -\frac{2\pi }{N}\mathrm{and}\pi +\frac{2\pi }{N}\mathrm{rad}$

• • [0239]  makes it possible to maximise the decrease in the overall size of the multi-interphase transformer,
  • [0240] dividing each winding e_1i connected to a phase of the power source into a plurality of windings e_1ik connected in series makes it possible to reduce the number of windings available to create cells, and thus to improve the possibility of approximation to an optimal configuration in which the difference in angular offsets x_i-x_j is equal to or very close to π rad.

[0241] Annex

[0242]

TABLE 1
C0	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11
V0	−V5	V5	−V10	V10	−V3	V3	−V8	V8	−V1	V1	−V6	V6	−V11	V11	−V4	V4	−V9	V9	−V2	V2	−V7	V7	−V0

[0243]

TABLE 4
C0	C6	C5	C11	C10	C4	C3	C9	C8	C2	C1	C7
V0	−V5	V11	−V6	V1	−V6	V0	−V7	V2	−V7	V1	−V8	V3	−V8	V2	−V9	V4	−V9	V3	−V10	V5	−V10	V4	−V11

[0244]

TABLE 5
C0	C6	C5	C11	C10	C4	C3	C9	C8	C2	C1	C7
V0	−V5	−V6	V11	V1	−V6	−V7	V0	V2	−V7	−V8	V1	V3	−V8	−V9	V2	V4	−V9	−V10	V3	V5	−V10	−V11	V4

[0245]

TABLE 6
Number of cells a multiple of 4
N	C0	C1	C2	C3	C4	C5	C6	C7	C8	C9
4	1	−2	−3	4	2	−3	−4	1
8	1	−4	−5	8	2	−5	−6	1	3	−6	−7	2	4	−7	−8	3
12	1	−6	−7	12	2	−7	−8	1	3	−8	−9	2	4	−9	−10	3	5	−10	−11	4
16	1	−8	−9	16	2	−9	−10	1	3	−10	−11	2	4	−11	−12	3	5	−12	−13	4
20	1	−10	−11	20	2	−11	−12	1	3	−12	−13	2	4	−13	−14	3	5	−14	−15	4
N	C10	C11	C12	C13	C14	C15	C16	C17	C18	C19
4
8
12	6	−11	−12	5
16	6	−13	−14	5	7	−14	−15	6	8	−15	−16	7
20	6	−15	−16	5	7	−16	−17	6	8	−17	−18	7	9	−18	−19	8	10	−19	−20	9

[0246]

TABLE 7
Odd number of cells
N	C0	C1	C2	C3	C4	C5	C6	C7	C8	C9
5	1	−3	−3	5	5	−2	−2	4	4	−1
7	1	−4	−4	7	7	−3	−3	6	6	−2	−2	5	5	−1
9	1	−5	−5	9	9	−4	−4	8	8	−3	−3	7	7	−2	−2	6	6	−1
11	1	−6	−6	11	11	−5	−5	10	10	−4	−4	9	9	−3	−3	8	8	−2	−2	7
13	1	−7	−7	13	13	−6	−6	12	12	−5	−5	11	11	−4	−4	10	10	−3	−3	9
15	1	−8	−8	15	15	−7	−7	14	14	−6	−6	13	13	−5	−5	12	12	−4	−4	11
17	1	−9	−9	17	17	−8	−8	16	16	−7	−7	15	15	−6	−6	14	14	−5	−5	13
19	1	−10	−10	19	19	−9	−9	18	18	−8	−8	17	17	−7	−7	16	16	−6	−6	15
	N	C10	C11	C12	C13	C14	C15	C16	C17	C18
5
7
9
11	7	−1
13	9	−2	−2	8	8	−1
15	11	−3	−3	10	10	−2	−2	9	9	−1
17	13	−4	−4	12	12	−3	−3	11	11	−2	−2	10	10	−1
19	15	−5	−5	14	14	−4	−4	13	13	−3	−3	12	12	−2	−2	11	11	−1

[0247]

TABLE 8
C0	C6	C5	C11	C10	C4	C3	C9	C8	C2	C1	C7
V0	−V5	−V6	V11	−V1	V6	V7	−V0	V2	−V7	−V8	V1	−V3	V8	V9	−V2	V4	−V9	−V10	V3	−V5	V10	V11	−V4