Abstract:
The invention concerns a machine forming a motor or generator including a stator and a rotor. The rotor is a passive rotor, consisting of two ferromagnetic discs whereof at least one is toothed. The stator includes a fixed polyphase field coil arranged in an air gap defined by the space provided between the two discs and generating a rotating magnetic field, and a field coil likewise fixed. The invention is applicable to a cylindrical rotary machine or to a machine with linear displacement.

Description:
BACKGROUND OF THE INVENTION 
     The present invention is concerned with rotating machines and relates more particularly to energised reluctance machines. 
     In the context of electromechanical energy storage, electric machines acting as motors and generators are generally conventional structures (synchronous with permanent magnets or asynchronous). 
     The combination of the storage function, properly so called, which is provided by an inertia wheel, with the motor/generator function is effected in two ways: 
     “decoupled”, where the motor/generator corresponds to a component of the most conventional topology, which is simply added on; 
     “integrated”, where the machine forms part of the inertia wheel. In that case too, the machines used are relatively conventional in design. 
     With regard to the topologies used, they may be as follows: 
     pure variable reluctance synchronous machine, with a radial magnetic field and coil in the air gap, guiding of which is effected by ball bearings assisted by magnetic bearings (for example the “Active Power” system); 
     synchronous machine having permanent magnets with a radial or axial field and with a coil in the air gap or with a coil in slots (various constructions); 
     asynchronous machine with a coil in slots and a solid rotor (for example the “Japanese Flywheel” system). 
     Those topologies do not have adjustable energisation, and certain ferromagnetic elements are fixed. Such topologies therefore have three major disadvantages: 
     the existence of losses on no-load operation (storage operation), leading to substantial self-discharge (except for the asynchronous machine and the pure variable reluctance machine); 
     low flexibility of adjustment of the energy exchanges by adjustment of the induction current; 
     poor efficiency at low charge. 
     SUMMARY OF THE INVENTION 
     The object of the invention is to provide an electromagnetic motor/generator for the electromechanical storage of the electrical energy with very high efficiency, especially with low self-discharge, which can be integrated with an inertia wheel. 
     To that end, the machine must simultaneously satisfy the following criteria: 
     small losses on no-load operation, that is to say operation without current, which must be manifested in an absence of magnetic losses in the iron and in the coil and very few mechanical losses, 
     no losses at the rotor despite the high rotational speeds (peripheral speed close to that of the inertia wheel), 
     no disturbance of the magnetic bearings by parasitic forces, 
     independent adjustment of the excitation flux for better control of the energy exchanges and a better power factor (minimum apparent power of the associated electronic converter), 
     good integration with an inertia wheel. 
     The invention accordingly relates to an electric machine forming a motor or generator, comprising a fixed part and a movable part, in which one of the fixed and movable parts is passive and comprises two elements of ferromagnetic material which define between them a regular interval, said elements being connected to one another by a joining element which is likewise made of ferromagnetic material, and the other of said fixed and movable parts comprises a multiphase armature coil, which generates a shifting field, as well as a centralized field coil supplied by a direct current, said armature coil and field coil being arranged in an air gap defined by the regular interval, characterized in that at least one of said elements of ferromagnetic material is provided with openings. 
     According to other features: 
     the rotor is a passive rotor comprising two coaxial elements of revolution which are made of ferromagnetic material and at least one of which is provided with openings, which elements are connected to one another by a joining element which is coaxial with respect to the two elements of revolution and is likewise made of ferromagnetic material, and the stator comprises, arranged in an air gap defined by the regular interval formed between the two elements of revolution, a fixed armature coil, which generates a rotating field, and a field coil, which is likewise fixed, 
     all the ferromagnetic elements are rotating, 
     the rotor is composed of two ferromagnetic disks, at least one of which is toothed, 
     the rotating disks are fixed on a shaft of ferromagnetic material, 
     the fixed armature coil and the fixed field coil are arranged in the space between the disks, 
     the armature coil is a multiphase coil whose phases are arranged in the same plane or according to superposed planes in the air gap formed by the rotating toothed disks, 
     the phases of the armature coil are formed by turns which are offset angularly and are distributed regularly along the periphery of the machine, 
     the field coil is arranged about the shaft and is surrounded by the armature coil, 
     the field coil is a fixed global coil in the form of a solenoid, 
     the rotor comprises a shaft surrounded by two coaxial cylinders of ferromagnetic material which are provided with teeth and cut-outs distributed regularly at one of their ends, and a joining flange of ferromagnetic material, 
     the rotor comprises a shaft surrounded by two cylinders of ferromagnetic material which are provided with holes distributed regularly at one of their ends, and a joining flange of ferromagnetic material, 
     the inner cylinder of ferromagnetic material is in one piece with the shaft of the rotor, 
     the fixed part comprises a rail of ferromagnetic material having lateral walls provided with openings distributed regularly over its length, and the movable part comprises a support of non-magnetic material carrying an armature coil and a field coil which are arranged in an air gap defined by the regular interval between the lateral walls, 
     the openings provided in the lateral walls of the rail are cut-outs separating teeth of said rail, 
     the openings provided in the lateral walls of the rail are holes formed at regular intervals in said lateral walls. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The invention will be better understood upon reading of the following description, which is given solely by way of example and makes reference to the attached drawings, in which: 
     FIG. 1 is a front cutaway view of the electric machine associated with its inertia wheel according to the invention; 
     FIG. 2 is a top view, partially cut away, of the machine of FIG. 1; 
     FIG. 3 is a schematic view of the armature coil of the machine according to the invention; 
     FIG. 4 is a linearly developed schematic view of an elementary structure of the machine according to the invention; 
     FIG. 5 is a diagram showing the variation of the excitation flux embraced by two turns of the structure of FIG. 4 as a function of the position of the rotor of the machine according to the invention; 
     FIGS. 6 a  and  6   b  are diagrams showing the energy conversion cycles of the machine according to the invention in two cases of the supply of sinusoidal armature current or armature current of rectangular wave form; 
     FIG. 7 is a diagram showing the permeances P 1  and P 2  viewed respectively by the two turns of the structure of FIG. 4 as a function of the position of the rotor; 
     FIG. 8 is a cutaway front view of a machine according to the invention of generally cylindrical shape; 
     FIG. 9 is a cross-sectional view of the machine of FIG. 8; 
     FIG. 10 is a cutaway front half-view of a variant of the machine of FIG. 8; 
     FIG. 11 is a developed schematic view of the machine of FIG. 10; 
     FIG. 12 is a schematic perspective view of a linear electric machine according to the invention; 
     FIG. 13 is a schematic view showing the flow of the current in the machine of FIG. 12; 
     FIG. 14 is a schematic cross-sectional view of a variant of the linear electric machine of FIG. 12; 
     FIG. 15 is an exploded schematic perspective view showing a particular embodiment of a flat coil which can be used in the machine of FIG. 1; 
     FIG. 16 is a partial view of FIG. 15 showing the structure of the flat coil of FIG. 15; 
     FIG. 17 is a partial cutaway view of an electric machine analogous to that of FIG. 1 but having a flat coil using the arrangement of FIG. 15; 
     FIG. 18 is a graph showing the determination of the width of the conductors located in the active part of a coil such as that of FIG. 16; and 
     FIG. 19 is a graph showing the appearance of the optimised magnetomotive force, which has been rendered as sinusoidal as possible thanks to the graphic determination of the width of the conductors of FIG.  18 . 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The machine shown in FIG. 1 comprises a casing  1  formed by a first flange  2  which supports a lower bearing  2   a  for the rotation of a shaft  3  and on which is mounted a tube  4  defining the lateral wall of the casing  1 , and a second flange  5  which forms the cover and supports an upper bearing  5   a.    
     Two toothed disks  8 ,  9  are mounted on the shaft  3 , each disk having four teeth  10  (FIG. 2) separated by cut-outs  10   a.    
     The lower disk  8  is integral with an inertia wheel  11  which, in this example, is of cylindrical shape. 
     In the space  12  formed between the disks  8 ,  9  there is arranged a distributed flat armature coil  13  surrounding a global field coil  14  in the form of a solenoid. 
     The field coil  14  is arranged between the two toothed disks  8 ,  9  in such a manner as to surround the shaft  3 , with mechanical clearance  15 . 
     The armature coil  13  is formed of turns which are offset angularly and distributed regularly. 
     As will be seen in FIG. 1, the three phases P 1 , P 2  and P 3  of the armature coil  13  are arranged according to three planes superposed on one another. 
     They may also be arranged in the same plane. 
     The armature coil is composed of a fixed multiphase coil, which in the present example is a three-phase coil. 
     The coil shown by way of example in FIG. 3 is a coil for a motor/generator having four pairs of poles. 
     That figure shows the three phases P 1 , P 2 , P 3  of the coil distributed uniformly along its periphery. 
     The principle of operation of the machine according to the invention is based on flux commutation. 
     The alternation of excitation flux viewed by a phase of the fixed armature coil is obtained from excitation by direct current and by the displacement of the pure reluctance circuit. 
     In order to illustrate that operation, a representative elementary structure shown in FIG.  4  and composed of magnetic teeth  20 , cut-outs  21  and elementary turns  22 ,  23  will be considered. 
     The magnetic air gap  12  determined by the mechanical clearance between the magnetic teeth  20 , increased by the height of the turns  22 ,  23 , is magnetised off-load by an excitation coil which, in FIG. 4, is shown by the magnetic potential difference  24  at the terminals of the two toothed rotors  8 ,  9  (FIG.  1 ). 
     The variation of the excitation flux in the turns  22  and  23  is shown in FIG.  5 . 
     Accordingly, the total excitation flux viewed by the two turns in series of the arrangement of FIG. 4, forming the armature coil, corresponds to the difference in the flux of the two turns. 
     FLUX=FLUX 1−FLUX 2 
     It is an alternating quantity with an average value of zero. 
     Starting from that variation in the total excitation flux, it is possible to show the extreme characteristics of the operating cycle of the device in the “flux-ampere-turns” plane. 
     The shape of FIGS. 6 a  and  6   b  is thus obtained, FIG. 6 a  showing the armature current supplying a phase (sinusoidal or in rectangular wave form) synchronized with the electromotive force of that same phase, FIG. 6 b  showing the variation in the flux as a function of the ampere-turns. The area of the cycle so described is equal to the energy converted per supply period. 
     With regard to the effect of variable reluctance, it is possible to show that it is negligible in this structure. 
     In fact, FIG. 7 shows the variation in the permeances of each elementary turn described above, as well as the total permeance, which represents the permeance of a phase of the armature coil and is equal to the sum of the elementary permeances. 
     For a sinusoidal distribution, as a function of the position of the rotor, of the elementary permeances, the total permeance is constant. 
     For a different distribution, it is not very variable, as is shown in FIG.  7  and as it has been possible to demonstrate by finite element analysis and during experiments. 
     It will be noted, however, that if the total permeance of a phase is not constant and is therefore dependent on the position of the rotor, there is an additional, so-called reluctance couple resulting from coupling of the induced magnetic field with the magnetic circuit. 
     The structure of the motor/generator according to the invention combines four fundamental aspects, namely: 
     a passive rotor which comprises neither a magnet nor a coil and which is therefore very robust mechanically and withstands the high speeds of rotation, 
     no parasitic forces, the system thus being compatible with magnetic bearings, 
     independent adjustment of the excitation flux, 
     all the ferromagnetic elements are rotating and therefore view a constant magnetic field, as a result of which the iron losses are zero. 
     Thanks to the invention, it is possible substantially to prolong the duration of energy storage and accordingly to envisage electromechanical storage with a very high degree of autonomy. 
     The cost of the device according to the invention is also very low owing to its great simplicity of construction and the use of inexpensive materials. 
     The device according to the invention additionally has the following features: 
     very few no-load losses because there is no magnetic no-load loss and only aerodynamic losses (which are nevertheless much reduced if the pressure is low owing to a partial vacuum), 
     sealability, 
     independent adjustment of the excitation, which gives a degree of additional freedom in the adjustment of the energy transfer, 
     negligible axial or radial parasitic forces between the movable parts and the fixed parts, 
     vibrations are negligible, 
     excellent compatibility with magnetic bearings, 
     ready integration of magnetic bearings, 
     topology can be integrated wholly or partially with an inertia wheel. 
     The fields of application of the device according to the invention relate to the electromechanical storage of electrical energy for: 
     consumers not connected to the supply system and supplied by renewable energies such as photo-voltaic energy, wind energy or the like; 
     consumers connected to the supply system, in order to smooth out consumption, ensure autonomy in case of power cuts and the possibility of producing electricity from renewable energy sources. 
     The invention makes it possible to have a storage device with a very large number of charge-discharge cycles with very low self-discharge, to recycle the product easily at the end of its life and, finally, a product of relatively simple construction composed of inexpensive materials such as steel, copper and others. 
     The machine shown in FIG. 8 is a variant of the machine described with reference to FIGS. 1 to  4 . 
     It comprises a hollow shaft  30  rotatably mounted by way of a bearing  32  in a fixed flange  34  of non-magnetic material carrying a multiphase armature coil  36  whose three phases P 1 , P 2 , P 3  are produced according to three coaxial cylinders surrounding the shaft  30 . 
     At the end of the armature coil  36  opposite the non-magnetic flange  34  there is arranged a fixed field coil  38 . 
     The armature coil  36  and the field coil  38  are arranged in the air gap defined by an interval  40  formed between two coaxial toothed cylinders of ferromagnetic material  41 ,  42  which are movable in rotation with the shaft  30  and are joined together by a flange  43  of ferromagnetic material. 
     The toothed cylinder  41  and the flange  43  are carried by the shaft  30 . According to a variant, the inner cylinder  43  can be in one piece with the shaft  30 , which is then made of magnetic material. 
     The toothed cylinders  41  and  42  have teeth  41   a ,  42   a , respectively, of which there are three in the present example, as shown in FIG. 9, which are separated by respective cut-outs  41   b ,  42   b.    
     The shaft  30 , the cylinders  41 ,  42  and the flange  43  form the rotor of the machine. 
     The armature coil  36  and the field coil  38  which are carried by the non-magnetic support  34  form the stator of the machine. 
     The machine shown in FIG. 10 is similar to that described with reference to FIGS. 8 and 9 except that, instead of having teeth and cut-outs, its cylinders  41 ,  42  have holes  41   c ,  42   c  distributed regularly at their periphery. The purpose of the holes  41   c ,  42   c  is the same as that of the cut-outs  41   b ,  42   b  in the embodiment of FIG.  8 . 
     This variant of the machine according to the invention is suitable for receiving a hollow cylindrical energy-storing element  44  made of material having high mechanical strength (HMS) of the HMS metal type or carbon fibres or alternatively glass fibres. 
     FIG. 11 shows a developed view of the machine of FIG.  10 . 
     That figure shows the circulation of the currents of the armature i 1 , i 2 , i 3  in the three phases P 1 , P 2 , P 3  of the armature coil which are offset relative to one another at regular intervals. 
     FIG. 12 is a schematic perspective view of a linear machine according to the invention. 
     That machine comprises a rail  50  of ferromagnetic material formed, for example, by a horizontal sole plate  52  and two lateral vertical plates  53 ,  54  provided with teeth  53   a ,  54   a  separated by cut-outs  53   b ,  54   b.    
     Assuming for the purposes of the description that the rail  50  is fixed, the machine additionally comprises a movable non-magnetic horizontal support  56  carrying a vertical armature coil  58  which extends into an air gap  60  formed by the interval between the lateral plates  53 ,  54  of the rail  50  and with which there is associated a field coil  62 . 
     The armature coil  58  is multiphase, three-phase in the present example, and, as is shown in FIG. 14, is surrounded by the field coil. 
     The assembly formed by the non-magnetic support  56 , the armature coil  58  and the field coil is movable in translation relative to the rail  50 . 
     It is, however, possible to envisage a machine in which the armature coil and the field coil, integral with their support, are fixed and the rail of ferromagnetic material is movable. 
     As will be seen in FIG. 13, the currents i 1 , i 2 , i 3  of the three phases P 1 , P 2 , P 3  of the armature behave in the linear machine of FIG. 12 in the same manner as in the rotating machine of FIG. 8 or  10 . 
     The machine shown diagrammatically in FIG. 14 is similar in all points with the linear machine described with reference to FIGS. 12 and 13 except that, instead of having teeth and cut-outs like the machine of FIG. 12, it has in the lateral ferromagnetic plates  53 ,  54  holes  53   c ,  54   c  arranged at regular intervals. 
     Accordingly, the invention is applicable both to machines having an axial field of disk-shaped topology, such as a machine used for the storage of energy, and to machines having a radial magnetic field, of the cylindrical type, or having a transverse field, such as linear displacement machines. 
     In those machines, all the ferromagnetic elements view a constant magnetic field. 
     In machines other than linear displacement machines, the armature coil and the field coil are fixed. 
     The armature coil and the field coil are located in the air gap defined in the space between the ferromagnetic elements formed by the ferromagnetic disks in the case of the disk-shaped machine, the ferromagnetic cylinders in the case of the cylindrical machine, and the ferromagnetic plates in the case of the linear machine. 
     The excitation flux of the machine is adjustable. 
     The armature coil is multiphase and generates a rotating field in a rotating machine or a shifting field in a linear machine. 
     The field coil is centralized and supplied by a direct current. 
     The armature coil is a principal element of the machine according to the invention. 
     The appearance of the magnetomotive force created by the armature and the appearance of the electromotive force induced by the inductor are dependent thereon. 
     The losses are therefore dependent, for a large part, on the type and form of the coil chosen. 
     A coil in the iron would exhibit an excessive level of losses of magnetic origin at the speeds required for electromechanical energy storage. 
     The machine, motor/generator according to the invention uses a coil having a non-magnetic support. 
     The solution having a wire coil such as that of the machine of FIG. 1 presents the problem for the machine in question, called the actuator, of the space requirement due to overlapping of the inner end windings. 
     Such a solution leads to under-use of the air gap volume. 
     In order to overcome the problem of overlapping, it is proposed to distribute the field coil and the armature coil on the same plate of insulating material, such as epoxy, which also provides mechanical strength, by using, for example, a printed circuit etching process. 
     The flat coil shown in FIG. 15 comprises a plate in the form of a disk  64  of insulating material, such as epoxy, which is provided with a central orifice  66  for the passage of an element for joining two toothed disks such as the magnetic disks  8 ,  9  of the machine of FIG.  1 . 
     On a first face  68  of the plate  64  there is produced, for example by a printed circuit etching process, a first part  70  of an armature coil whose various poles  72  are arranged flat on the plate  64  and distributed at regular angular intervals over said plate. 
     On a second face  74  of the plate  64  there is produced, by the same process, a second part  76  of the armature coil whose poles  78  are likewise arranged flat on the plate  64  with the same angular offset as that of the poles  72  of the first part  70  of the armature coil. 
     Each of the poles  72 ,  78  of a face of the assembly or pancake coil so obtained is wound in the opposite direction to two adjacent poles, which creates magnetic pole alternation. 
     The poles  72 ,  78  created by the two faces of the pancake coil are superposed. They are therefore wound in the same direction. 
     Furthermore, the internal connections of the poles  72  of one face  68  which are provided by the terminals  79  are connected to the internal connections or terminals  80  of the corresponding poles  78  of the other face  74  by way of through-holes  81  formed in the plate  64  of insulating material and arranged at the periphery thereof opposite the terminals  79 ,  80  of the poles  72 ,  78  of the two armature coil parts  70 ,  76 . 
     The number of poles  72 ,  78  of a pancake coil is equal to twice the number of teeth of a rotor disk such as the disks  8 ,  9  of the machine of FIG.  1 . 
     The armature coil so formed has two external terminals  82  connected to two adjacent poles  72  of the first part  70 . Each of the parts  70  and  76  of the armature coil has a circular configuration and comprises a central zone in which a field coil part  83 ,  84  is produced by the same printed circuit etching technique. 
     Each field coil part  83 ,  84  comprises an external terminal  86 ,  88  and an internal terminal  90 ,  92 . The internal terminals  90 ,  92  are connected together by way of a through-orifice  94  formed in the insulating plate  64 . 
     As is shown in the partial view of FIG. 16, in which there will be seen an example of a face of a pancake coil such as that of FIG. 15, a pole such as a pole  72  of the armature coil is formed by a flat-rolled printed conductor which comprises internal peripheral portions  93  and external peripheral portions  94  of constant cross-section, and radial portions  95  of variable cross-section which narrow from the periphery to the centre. 
     Furthermore, the widths of the portions  95  of the printed conductor, radial or transverse to the relative displacement between the fixed part and the movable part of the machine, diminish starting from the middle of a pole to its edges. 
     Such an optimised distribution of the conductors allows a magnetomotive force that is as sinusoidal as possible to be obtained. 
     It is, however, possible to envisage other optimisation criteria. 
     The cross-section of each of the conductors located in the active part, that is to say the part located beneath the toothed zone of the rotor disks, is dependent on the shape of the magnetomotive force which is desired to be obtained. The determination of that cross-section, for a given radius, can be carried out starting from a simple graphic analysis. 
     For a magnetomotive force (m.m.f.) of the desired shape, such as that shown by dotted lines in FIG. 18, and for a constant track thickness, the width of each track is obtained as follows: 
     i. the intersection of the curve representing the desired magnetomotive force with the integer levels of the ampere-turns determines the value Δθ k , 
     ii. the width of the k th  track, denoted Δθ track k , is obtained by subtracting from Δθ k  the value of the difference Δθ i  corresponding to the distance necessary for electrical insulation between two tracks. 
     
       
         Δθ track k =Δθ k −Δθ i    
       
     
     By way of example, FIG. 18 shows, in the case of rotating machines, how to obtain the cross-section of the conductors forming a pole in the case where each pole comprises three turns and where the desired magnetomotive force is triangular. The top of the figure shows physically a face of the coil seen in section and for a given radius. 
     In that example, the conductors  95  in the form of copper tracks are arranged on the surface of a non-magnetic plate  64  of epoxy, for example, with a constant angular pitch Δθ i , necessary for electrical insulation between tracks. 
     In that precise example, the resulting track width is then constant. 
     In the case where the magnetomotive force is not triangular, the width of the variable tracks is determined using the method described above. Thus, the distribution of the conductors of the motor/generator in question, one face of a pancake coil of which is shown in FIG. 16, has been optimised to obtain, starting from 13 turns per pole and per phase, a magnetomotive force shown in FIG. 19 that is as sinusoidal as possible. 
     FIG. 17 is a partial cutaway view on an enlarged scale of a machine analogous to that of FIG.  1 . 
     FIG. 17 shows the shaft  96  carrying two toothed disks  98 ,  99  which are integral with a hub  100  keyed on the shaft, and which are provided with openings  100   a.    
     In the space  101  formed between the toothed disks  98  and  99  there is arranged a three-phase armature coil  102  whose three phases P 1 , P 2 , P 3  are each formed of a plurality of respective pancake coils  104   a ,  104   b ,  104   c  such as that described with reference to FIG. 15, which are stacked on one another with the interposition of a layer  106  of electrical insulation between the pancake coils  104   a ,  104   b ,  104   c.    
     In FIG. 17, only the outer pancake coils of each stack of phases P 1 , P 2 , P 3  are shown. 
     Of course, the coil may comprise only one pancake coil per phase. 
     In the case of a monophase machine, it comprises at least one pancake coil of the above-mentioned type. 
     The pancake coils  104   a ,  104   b ,  104   c  of each phase P 1 , P 2 , P 3  each comprise an armature coil  70 ,  76  and a field coil  83 ,  84  each formed of two parts printed on the two faces of a disk  64  of insulating material in the manner described with reference to FIG.  15 . 
     The two parts  70 ,  76  of each armature coil are connected to one another by through-conductors  108 . The armature coils of the pancake coils  104   a  forming the same phase P 1  are connected to one another in series by conductors  110   a.    
     Likewise, the armature coils of the pancake coils  104   b  and  104   c  forming the phases P 2  and P 3 , respectively, are connected to one another by conductors  110   b  and  110   c.    
     The field coils  83 ,  84  of the assembly of pancake coils  104   a  to  104   c  forming the three phases are connected in series by conductors  112  and constitute a centralized field coil  114 . 
     The number of printed circuit pancake coils used to produce the phases P 1 , P 2 , P 3  is chosen so that the resulting coil occupies a maximum volume between the toothed disks  98 ,  99 . 
     Thanks to the use and the stacking of a plurality of double-faced printed circuits of the type described with reference to FIG. 15, overlapping of the end windings is avoided. 
     In the case of a three-phase displacement machine, the phases P 1 , P 2 , P 3  are offset relative to one another by a value corresponding to two thirds of a pole such as one of the poles  72  of the coil of FIG.  15 . 
     The type of coil just described with reference to FIGS. 15 to  17  can be transposed directly, by simple development, to linear machines such as those described with reference to FIGS. 11 to  14 . 
     In the case of a linear machine, the offset between poles is effected according to the direction of relative displacement of the fixed part and the movable part. 
     The technology described with reference to FIGS. 15 to  17  allows the following advantages to be obtained: 
     a field coil and an armature coil arranged on the same non-magnetic element which is not a conductor of electricity (epoxy, for example) and which provides mechanical strength; 
     a virtually sinusoidal magnetomotive force obtained in a very simple manner by the use of tracks of variable cross-sections, which is advantageous for minimising magnetic losses in disks of iron; 
     better use of the coilable space; 
     the possibility of increasing the thermal exchange surfaces with the exterior by increasing the cross-section of the end windings; 
     a reduction in Joule losses by increasing the cross-section of the end windings; 
     increased simplicity of production, which is advantageous for automation and for reducing production costs.