Improving the Efficiency of the Axial Flux Machine with Hybrid Excitation

: This paper discusses the construction and operating principle of an axial flux electric machine with hybrid excitation. Based on computer simulations using the Finite Element Method, an analysis was conducted with changes in the geometry of the magnetic circuit, which involves the rotation of the rotor disks relative to each other on the operating parameters of the machine. Both the generator state of operation, in the meaning of analyzing the induced voltage (adjustment at − 11% ÷ +64%) and the cogging torque, and the motor state of operation, in the meaning of analyzing the ripple of the electromagnetic torque (possible reduction by almost 30%), were examined. The article concludes with observations on how the change in the angle of the rotor disks affects the efficiency of the disk machine with axial flux and hybrid excitation.


Introduction
Following the improvement in the parameters of permanent magnets that generate strong magnetic fields, scientific centers began searching for design solutions for electric machines that would be characterized by high energy efficiency and better dynamics and reliability but also good power-to-weight ratio properties.The two basic designs of electromechanical energy converters are machines with radial flux direction (Figure 1a) and axial flux direction (Figure 1b) [1][2][3][4][5].
Electronics 2024, 13, x FOR PEER REVIEW 2 of 12 currents in the stator winding.This operating range is referred to as constant power operation because as the rotational speed further increases, the power remains at the same level, but the torque on the machine's shaft decreases.The further weakening of the field can be achieved by introducing an additional, adjustable excitation source.Figure 2 presents the mechanical characteristic of a machine with hybrid excitation, illustrating the dependence of the electromagnetic torque (T-red color) voltage on the windings (Ugreen color) and the dependence of electric power (P-blue color) on the rotational speed n.The possibility of modifying the electromagnetic torque through techniques for enhancing or weakening the magnetic field in the machine using an additional excitation source is indicated in orange color.An additional excitation source can be installed in series or in parallel with the primary excitation source [14][15][16][17].Due to significantly greater reliability, radial flux solutions are most commonly used in the industry.However, machines with axial flux are characterized by better power-toweight and torque-to-weight ratios [6][7][8][9][10][11].They are also not without flaws, including a high cogging torque and torque ripple, which many research centers around the world are working on [12,13].
Modern electric machines with permanent magnets are also required to operate at a wide range of rotational speeds.The speed control of the machine below the base speed n b is achieved by regulating voltage at the maximum allowable current for the stator windings.This operating range is known as constant torque operation.Once the nominal values of voltage and currents are reached, the machine rotates at the base speed n b .Operation above this speed is only possible by weakening the main magnetic flux in the machine, which can be achieved through appropriate techniques of controlling the machines' currents in the stator winding.This operating range is referred to as constant power operation because as the rotational speed further increases, the power remains at the same level, but the torque on the machine's shaft decreases.The further weakening of the field can be achieved by introducing an additional, adjustable excitation source.Figure 2 presents the mechanical characteristic of a machine with hybrid excitation, illustrating the dependence of the electromagnetic torque (T-red color) voltage on the windings (U-green color) and the dependence of electric power (P-blue color) on the rotational speed n.The possibility of modifying the electromagnetic torque through techniques for enhancing or weakening the magnetic field in the machine using an additional excitation source is indicated in orange color.An additional excitation source can be installed in series or in parallel with the primary excitation source [14][15][16][17].
Electronics 2024, 13, x FOR PEER REVIEW 2 of 12 currents in the stator winding.This operating range is referred to as constant power operation because as the rotational speed further increases, the power remains at the same level, but the torque on the machine's shaft decreases.The further weakening of the field can be achieved by introducing an additional, adjustable excitation source.Figure 2 presents the mechanical characteristic of a machine with hybrid excitation, illustrating the dependence of the electromagnetic torque (T-red color) voltage on the windings (Ugreen color) and the dependence of electric power (P-blue color) on the rotational speed n.The possibility of modifying the electromagnetic torque through techniques for enhancing or weakening the magnetic field in the machine using an additional excitation source is indicated in orange color.An additional excitation source can be installed in series or in parallel with the primary excitation source [14][15][16][17].Motors that operate in severe working conditions must be protected because they are susceptible to high temperatures, fluctuations, and disturbances.With the phase current, the temperature distributions in stators would change.Furthermore, both the temperature and the maximum temperature increase when the excitation current is exerted.The temperature distributions in armature windings and field windings would likewise be impacted by the excitation current.Therefore, the maximum temperature of the armature windings increases with an increase in field current.The heat distributions on the copper windings of the armature are influenced by the excitation field.PM demagnetization may occur as a result of the excitation windings' temperature increase.Hence, it is important to address PM demagnetization brought on by high temperatures.More research is required on stator heat dissipation and the demagnetization hazards brought on by the Motors that operate in severe working conditions must be protected because they are susceptible to high temperatures, fluctuations, and disturbances.With the phase current, the temperature distributions in stators would change.Furthermore, both the temperature and the maximum temperature increase when the excitation current is exerted.The temperature distributions in armature windings and field windings would likewise be impacted by the excitation current.Therefore, the maximum temperature of the armature windings increases with an increase in field current.The heat distributions on the copper windings of the armature are influenced by the excitation field.PM demagnetization may occur as a result of the excitation windings' temperature increase.Hence, it is important to address PM demagnetization brought on by high temperatures.More research is required on stator heat dissipation and the demagnetization hazards brought on by the excitation field.In order to reduce the possibility of demagnetization, a comprehensive thermal study must be performed, which could come under the purview of future studies [18].
It is crucial to estimate the heat flux and temperature distribution inside the analyzed machine.In the case of axial heat transmission, the radial structure of the axial machine should be examined.Through simulations, the heat flux and temperature distri-Electronics 2024, 13, 2219 3 of 12 bution within the axial machine-specifically, in the rotor-were investigated.Calculations unambiguously show how heat moves throughout the axial machine and evaluate the temperatures brought on by thermally conductive components.To guarantee that the heat is delivered evenly across the various axial machine zones, thermal analysis is crucial.Thermal examinations reveal causes of loss, including bearing losses, as well as different heat transmission systems, such conduction and convection heat.Therefore, to ensure that the axial machine operates smoothly and has a longer lifespan, an appropriate cooling system must be installed.Typically, a room temperature water cooling system is used to restrict the highest temperature.The purpose of such a water cooling system is to maintain the outside frame's center section at a temperature that is almost identical to the ambient temperature.As a result, the cooling system should be developed in order to restrict the increase in temperature of the rotor's surface as well as its hot spot.Future study may focus on determining the proper cooling system for the axial flux machine with hybrid excitation, as this is crucial to improving the machine's performance [19,20].
The temperature of the electromechanical energy converter, the operating parameters, and their regulation all depend on the applied control method.In the case of machines with an additional excitation system, control is designed individually for a specific solution and takes into account many additional factors related to the design of the machine's magnetic circuit, as well as its bearings and cooling system [21][22][23][24][25].
The primary objective of this publication is to present an innovative design and demonstrate a new approach to the problem, such as torque ripple, and one of the methods for its elimination.

Description of the Machine
The electromechanical energy converter design under consideration is a disc type machine with one double-wound stator and two external discs connected via a ferromagnetic bushing, forming the rotor of the machine (Figure 3) [26,27].
excitation field.In order to reduce the possibility of demagnetization, a comprehensive thermal study must be performed, which could come under the purview of future studies [18].
It is crucial to estimate the heat flux and temperature distribution inside the analyzed machine.In the case of axial heat transmission, the radial structure of the axial machine should be examined.Through simulations, the heat flux and temperature distribution within the axial machine-specifically, in the rotor-were investigated.Calculations unambiguously show how heat moves throughout the axial machine and evaluate the temperatures brought on by thermally conductive components.To guarantee that the heat is delivered evenly across the various axial machine zones, thermal analysis is crucial.Thermal examinations reveal causes of loss, including bearing losses, as well as different heat transmission systems, such conduction and convection heat.Therefore, to ensure that the axial machine operates smoothly and has a longer lifespan, an appropriate cooling system must be installed.Typically, a room temperature water cooling system is used to restrict the highest temperature.The purpose of such a water cooling system is to maintain the outside frame's center section at a temperature that is almost identical to the ambient temperature.As a result, the cooling system should be developed in order to restrict the increase in temperature of the rotor's surface as well as its hot spot.Future study may focus on determining the proper cooling system for the axial flux machine with hybrid excitation, as this is crucial to improving the machine's performance [19,20].
The temperature of the electromechanical energy converter, the operating parameters, and their regulation all depend on the applied control method.In the case of machines with an additional excitation system, control is designed individually for a specific solution and takes into account many additional factors related to the design of the machine's magnetic circuit, as well as its bearings and cooling system [21][22][23][24][25].
The primary objective of this publication is to present an innovative design and demonstrate a new approach to the problem, such as torque ripple, and one of the methods for its elimination.

Description of the Machine
The electromechanical energy converter design under consideration is a disc type machine with one double-wound stator and two external discs connected via a ferromagnetic bushing, forming the rotor of the machine (Figure 3) [26,27].On each of the two rotor discs (1), permanent magnets (2) and iron poles (3) are alternately mounted, forming together six pairs of poles.The rotor discs and iron poles are made of solid steel.The permanent magnets are NeFeB with higher temperature coefficient (N38H) rectangular prisms with dimensions of 50 × 50 × 12 mm.The iron pole has a trapezoidal cross-section with a height of 15 mm.The discs are positioned such that the permanent magnets on one of the rotor discs are aligned opposite to the permanent magnets mounted on the other rotor disc.The discs are connected to each other by a ferromagnetic bushing (4).For the rotor discs, iron poles, and bushings, an alloy steel type 1010 was adopted in the model.
On the stator core (5), on the outer sides, there are two three-band windings (6) that are initially connected in series, which are wound in such a way that the winding of phase A is opposite to phase A. The stator yoke is constructed from toroidally wound M400-50A sheet steel.The stator winding consists of copper wire with a diameter of 0.7 mm.Each phase comprises 12 coils (6 on each side of the stator), each wound with two parallel wires and with 150 turns.Inside the stator, between the internal terminal connections of the windings, there is an additional excitation source in the form of an electromagnetic coil (7).The additional coil consists of 500 turns wound with copper wire of 1 mm diameter.The windings of the additional coil, just like the stator windings, have been led outside in a brushless manner.The main mechanical and geometric parameters are shown in Table 1.In the basic operating state (without powering the additional electromagnetic coil in the stator, I DC = 0 A) (Figure 4a), the magnetic field from the permanent magnets passes through the stator core and closes through the ferromagnetic bushing connecting the rotor discs (Φ PM/PM ).Part of the flux from the permanent magnets closes through the stator core and the iron pole, passing twice through the same air gap (Φ PM/IP ).Powering the additional coil with a current I DC = −5 A (Figure 4b) causes a generation of a magnetic flux in the same direction as the flux from the permanent magnets (Φ DC/PM ).The result is a higher saturation of the ferromagnetic bushing and counteraction to the flux from the permanent magnets in the air gap under the iron pole (Φ DC/IP ), i.e., a net reduction in the magnetic flux in the air gap.This operating state of the machine is called de-excitation.
Powering the additional coil with a current I DC = 5 A (Figure 4c) causes the generation of a magnetic flux in the opposite direction to the flux from the permanent magnets (Φ DC/PM ).The result is an enhancement in the flux from the permanent magnets in the air gap under the iron pole (Φ DC/IP ), i.e., a net increase in the magnetic flux in the air gap.This operating state of the machine is called over-excitation.The flow of magnetic fluxes in the machine is presented in Figure 4.
The aim of this publication is to analyze the parameters and efficiency of the machine when changing the geometry of the magnetic circuit, which involves rotating the rotor discs relative to each other.This is possible because of the rigid connection of the rotor discs with each other by a ferromagnetic bushing.The discs were rotated by an angle α = 0÷30 • mechanical, which is equivalent to shifting by half of a magnetic pole.It was assumed that increasing the angle α beyond 30 • would cause symmetrical changes.The shifts of the rotor by three characteristic angles α = 0 • , α = 15 • , and α = 30 • are presented in Figure 5.The aim of this publication is to analyze the parameters and efficiency of the machine when changing the geometry of the magnetic circuit, which involves rotating the rotor discs relative to each other.This is possible because of the rigid connection of the rotor discs with each other by a ferromagnetic bushing.The discs were rotated by an angle α = 0÷30° mechanical, which is equivalent to shifting by half of a magnetic pole.It was assumed that increasing the angle α beyond 30° would cause symmetrical changes.The shifts of the rotor by three characteristic angles α = 0°, α = 15°, and α = 30° are presented in Figure 5.The aim of this publication is to analyze the parameters and efficiency of the machine when changing the geometry of the magnetic circuit, which involves rotating the rotor discs relative to each other.This is possible because of the rigid connection of the rotor discs with each other by a ferromagnetic bushing.The discs were rotated by an angle α = 0÷30° mechanical, which is equivalent to shifting by half of a magnetic pole.It was assumed that increasing the angle α beyond 30° would cause symmetrical changes.The shifts of the rotor by three characteristic angles α = 0°, α = 15°, and α = 30° are presented in Figure 5.

Results of Simulation Tests
The simulation studies were conducted using the Finite Element Method in the ANSYS Maxwell environment.To reduce the computation time in the simulation studies, onesixth of the model was analyzed, and the obtained results were calculated to represent the parameters of the entire machine.In the model, boundary conditions were adopted whereby the magnetic field at the model's cut planes satisfies symmetry conditions, and the magnetic field does not extend beyond the analyzed area in the space around the model.All of the typical relevant simplifications have been performed.The slot opening cannot be more simplified, because the results of the magnetic field induction will be inconsistent with the theory.The distribution of magnetic field induction in the machine depending on the angular displacement between the rotor discs with the auxiliary coil de-energized is presented in Figure 6.
one-sixth of the model was analyzed, and the obtained results were calculated to represent the parameters of the entire machine.In the model, boundary conditions were adopted whereby the magnetic field at the model's cut planes satisfies symmetry conditions, and the magnetic field does not extend beyond the analyzed area in the space around the model.All of the typical relevant simplifications have been performed.The slot opening cannot be more simplified, because the results of the magnetic field induction will be inconsistent with the theory.The distribution of magnetic field induction in the machine depending on the angular displacement between the rotor discs with the auxiliary coil deenergized is presented in Figure 6.The quality of the generated mesh used in the FEM program directly affects the simulation's sensitivity.For this reason, the ANSYS program's option was utilized to generate a grid that is tailored to the model, and its additional densification slightly affects the results' quality but significantly impacts the computation time.Manually reducing the number of mesh nodes can significantly affect the measurement results.Additionally, increasing the number of finite elements also did not affect the results significantly-only the computation time increased.Below, Table 2 is presented, showing the number of measurement nodes in the main parts of the machine for different angular displacements between the rotor discs.One of the generated meshes, for an offset angle between the rotor discs of  = 30°, is shown in Figure 7.The quality of the generated mesh used in the FEM program directly affects the simulation's sensitivity.For this reason, the ANSYS program's option was utilized to generate a grid that is tailored to the model, and its additional densification slightly affects the results' quality but significantly impacts the computation time.Manually reducing the number of mesh nodes can significantly affect the measurement results.Additionally, increasing the number of finite elements also did not affect the results significantly-only the computation time increased.Below, Table 2 is presented, showing the number of measurement nodes in the main parts of the machine for different angular displacements between the rotor discs.One of the generated meshes, for an offset angle between the rotor discs of α = 30 • , is shown in Figure 7.The first set of studies involved the simulation of Electromotive Force (EMF) induced in the stator windings.The studies were performed in an unloaded state, and the machine rotated at a speed of 1000 rpm.The studies were carried out for three forced currents in the windings of the additional DC coil: I DC = 0 A, I DC = −5 A (which is the state of de-excitation of the machine), and I DC = 5 A (which is the state of over-excitation of the machine).The relationship between the effective value of EMF and the angle of disc displacement α is shown in Figure 8.The first set of studies involved the simulation of Electromotive Force (EMF) induced in the stator windings.The studies were performed in an unloaded state, and the machine rotated at a speed of 1000 rpm.The studies were carried out for three forced currents in the windings of the additional DC  = 0 A,  = −5 A (which is the state of deexcitation of the machine), and  = 5 A (which is the state of over-excitation of the machine).The relationship between the effective value of EMF and the angle of disc displacement α is shown in Figure 8.It can be observed that for  = 0 A, as the angle  increases, the effective value of EMF decreases, and at  = 30°, it equals 0 V.This happens because when the discs are rotated relative to each other, the main magnetic flux induces EMF in the windings on one side of the stator with an opposite direction compared to the EMF induced in the winding on the other side of the stator.For the angle  = 30°, the EMF induced in both windings has the same value, and since they are of opposite directions, the sum of the EMFs equals 0 V.To confirm these assumptions, the studies were repeated with a change in the stator connections, such that the winding on the other side of the stator was connected in the reverse direction as per Figure 9.The results of the simulation studies are presented in Figure 10.The first set of studies involved the simulation of Electromotive Force (EMF) induced in the stator windings.The studies were performed in an unloaded state, and the machine rotated at a speed of 1000 rpm.The studies were carried out for three forced currents in the windings of the additional DC coil:  = 0 A,  = −5 A (which is the state of deexcitation of the machine), and  = 5 A (which is the state of over-excitation of the machine).The relationship between the effective value of EMF and the angle of disc displacement α is shown in Figure 8.It can be observed that for  = 0 A, as the angle  increases, the effective value of EMF decreases, and at  = 30°, it equals 0 V.This happens because when the discs are rotated relative to each other, the main magnetic flux induces EMF in the windings on one side of the stator with an opposite direction compared to the EMF induced in the winding on the other side of the stator.For the angle  = 30°, the EMF induced in both windings has the same value, and since they are of opposite directions, the sum of the EMFs equals 0 V.To confirm these assumptions, the studies were repeated with a change in the stator connections, such that the winding on the other side of the stator was connected in the reverse direction as per Figure 9.The results of the simulation studies are presented in Figure 10.It can be observed that for I DC = 0 A, as the angle α increases, the effective value of EMF decreases, and at α = 30 • , it equals 0 V.This happens because when the discs are rotated relative to each other, the main magnetic flux induces EMF in the windings on one side of the stator with an opposite direction compared to the EMF induced in the winding on the other side of the stator.For the angle α = 30 • , the EMF induced in both windings has the same value, and since they are of opposite directions, the sum of the EMFs equals 0 V.To confirm these assumptions, the studies were repeated with a change in the stator connections, such that the winding on the other side of the stator was connected in the reverse direction as per Figure 9.The results of the simulation studies are presented in Figure 10.From the data presented in Figures 8 and 10, it can be observed that after powering the additional coil with a current of I DC = 5 A, the magnetic field is stronger, and the EMF increases by about 53%, regardless of the displacement of the discs relative to each other.Powering the additional coil in the opposite direction with I DC = −5 A results in a weakening of the magnetic field in the air gap, which leads to a reduction in the value of the induced EMF by about 11%.The ability to regulate the EMF within a range of approximately 65% is a very good result, especially considering that it does not depend on the angle of displacement of the discs relative to each other.From the data presented in Figures 8 and 10, it can be observed that after powering the additional coil with a current of  = 5 A, the magnetic field is stronger, and the EMF increases by about 53%, regardless of the displacement of the discs relative to each other.Powering the additional coil in the opposite direction with  = −5 A results in a weakening of the magnetic field in the air gap, which leads to a reduction in the value of the induced EMF by about 11%.The ability to regulate the EMF within a range of approximately 65% is a very good result, especially considering that it does not depend on the angle of displacement of the discs relative to each other.
The next study was to check how the displacement of the discs relative to each other affects the value of the cogging torque.The result of the simulation is presented in Figure 11.It can be observed that for  = 0 A, the cogging torque changes from about 4 Nm to about 1.5 Nm.At certain angles, there was a decrease in cogging torque.This trend was repeatable every 5° ; so, for  = 0° ,  = 10° ,  = 20° , and  = 30° , the cogging torque had the highest values, while for  = 5°,  = 15°, and  = 25°, the cogging torque had the lowest values.For a direct current in the additional  = −5 A in the de-excitation state of the machine, the change in cogging torque was small, and the cogging torque increased by about 15%.The cogging torque significantly increased in the over-excitation state of the machine  = 5 A. For the angle  = 0°, it increased to about 25 Nm, which is more than 600%.The same was true for  = 10°,  = 20°, and  = 30°.It can be observed that for a wide range of angles α, the increase in cogging torque was around 300%, which means that by slightly decreasing the angle of displacement of the discs relative to each other, the cogging torque can be reduced by about 50% even in the over-excitation state of the machine.The machine torque ripple arises mainly as an effect of the cogging torque.It is very important to reduce it because it will generate not only noise but also friction, so it will decrease the efficiency and lifetime of the machine.The cogging torque The next study was to check how the displacement of the discs relative to each other affects the value of the cogging torque.The result of the simulation is presented in Figure 11.It can be observed that for I DC = 0 A, the cogging torque changes from about 4 Nm to about 1.5 Nm.At certain angles, there was a decrease in cogging torque.This trend was repeatable every 5 • ; so, for α = 0 • , α = 10 • , α = 20 • , and α = 30 • , the cogging torque had the highest values, while for α = 5 • , α = 15 • , and α = 25 • , the cogging torque had the lowest values.For a direct current in the additional I DC = −5 A in the de-excitation state of the machine, the change in cogging torque was small, and the cogging torque increased by about 15%.The cogging torque significantly increased in the over-excitation state of the machine I DC = 5 A. For the angle α = 0 • , it increased to about 25 Nm, which is more than 600%.The same was true for α = 10 • , α = 20 • , and α = 30 • .It can be observed that for a wide range of angles α, the increase in cogging torque was around 300%, which means that by slightly decreasing the angle of displacement of the discs relative to each other, the cogging torque can be reduced by about 50% even in the over-excitation state of the machine.The machine torque ripple arises mainly as an effect of the cogging torque.It is very important to reduce it because it will generate not only noise but also friction, so it will decrease the efficiency and lifetime of the machine.The cogging torque performance is mainly dependent on the stator teeth and rotor poles' shapes, and it is particularly sensitive in hybrid excited machines.Cogging torque itself is an undesirable phenomenon, so efforts should be made to minimize it [28][29][30].
Electronics 2024, 13, x FOR PEER REVIEW 9 of 12 performance is mainly dependent on the stator teeth and rotor poles' shapes, and it is particularly sensitive in hybrid excited machines.Cogging torque itself is an undesirable phenomenon, so efforts should be made to minimize it [28][29][30].The last simulation study was the impact of the displacement of the discs relative to each other on the pulsations of the electromagnetic torque in the machine.The machine operated in motor mode and was powered by a sinusoidal current with a density of 5 A/mm , which was supposed to ensure that heat generation during operation remained The last simulation study was the impact of the displacement of the discs relative to each other on the pulsations of the electromagnetic torque in the machine.The machine operated in motor mode and was powered by a sinusoidal current with a density of 5 A/mm 2 , which was supposed to ensure that heat generation during operation remained at a safe level, eliminating the need for a cooling system.The studies were conducted for displacement angles α = 0 • and α = 4 • .The waveforms of the electromagnetic torque are presented in Figure 12, and the characteristic values are listed in Tables 3 and 4. The last simulation study was the impact of the displacement of the discs relative to each other on the pulsations of the electromagnetic torque in the machine.The machine operated in motor mode and was powered by a sinusoidal current with a density of 5 A/mm , which was supposed to ensure that heat generation during operation remained at a safe level, eliminating the need for a system.The studies were conducted for displacement angles  = 0° and  = 4° .The waveforms of the electromagnetic torque are presented in Figure 12, and the characteristic values are listed in Tables 3 and 4.
Powering the additional coil in the stator circuit affects the average values of the electromagnetic torque regardless of the disc displacement angle.When the coil is not powered ( = 0 A) and  = 0°, the average value of the electromagnetic torque is approximately 121 Nm.Powering the additional coil in the stator circuit affects the average values of the electromagnetic torque regardless of the disc displacement angle.When the coil is not powered (I DC = 0 A) and α = 0 • , the average value of the electromagnetic torque is approximately 121 Nm.
After powering the coil and de-exciting the machine, this value decreases to about 114 Nm (I DC = −5 A), a reduction of about 5%, while in the over-excitation state of the machine (I DC = 5 A), the average value of the electromagnetic torque increases to about 161 Nm, an increase of about 33%.This means that the additional DC coil installed in the stator circuit allows for the regulation of the electromagnetic torque within a range of about 40%.Changing the displacement angle between the discs by 4 degrees (α = 4 • ) had a minor effect on the average values of the electromagnetic torque, changing it by about 2%.
The change in the angle of displacement between the disks significantly affected the pulsations of the electromagnetic torque, which was calculated using the following expression: The change in the angle α resulted in a reduction in pulsations from 68% to 40% (for I DC = 5 A), from 31% to 23% (for I DC = 0 A), and from 28% to 20% (for I DC = −5 A).

Conclusions
Based on the conducted research, it can be stated that the analyzed disc machine with permanent magnets and a hybrid excitation flux has very good capabilities for regulating its parameters.The de-excitation state of the machine allows for a reduction in the induced voltage by about 11% and the electromagnetic torque by about 5%.However, the excited state of the machine allows for an increase in EMF by about 65% and the electromagnetic torque by about 33%.A disadvantage of this design is the high value of the cogging torque in the excited state, which is over 600% higher compared to the machine with an unenergized DC coil.One of the methods to reduce the cogging torque and thus also the pulsations of the electromagnetic torque involves the displacement of the magnetic poles mounted on the rotor discs relative to each other.Simulation studies have shown that a mechanical displacement of the disks by α = 4 • will bring many benefits, including the reduction in the cogging torque by about 50% (even in the over-excitation state of the machine) and a reduction in electromagnetic torque pulsations from 68% to 40% (for I DC = 5 A), from 31% to 23% (for I DC = 0 A), and from 28% to 20% (for I DC = −5 A), increasing the machine's efficiency, with only a minor decrease in its basic parameters (average values of the EMF by about 4% and the electromagnetic torque by about 2%).
The benefits advocating for further research on this complex structure include a wide range of adjustments in the electrical and mechanical parameters of the machine.A significant advantage of the design modification, which involves shifting the rotor discs relative to each other, is the reduction in electromagnetic torque ripples.Conducted simulation results give adequate information to modify a prototype of an existing machine, build dedicated experimental stands, and conduct experimental tests to confirm obtained results.

Figure 2 .
Figure 2. Mechanical characteristics of the machine with hybrid excitation.

Figure 3 .
Figure 3. Design of machine with axial flux and hybrid excitation.

Figure 4 .
Figure 4. Distribution of magnetic fluxes in the machine for different currents in an additional electromagnetic coil: (a)  = 0 A, (b)  = −5 A, (c)  = 5 A.

Figure 5 .
Figure 5. Changing the angle between the rotor discs.

Figure 4 .Figure 4 .
Figure 4. Distribution of magnetic fluxes in the machine for different currents in an additional electromagnetic coil: (a) I DC = 0 A, (b) I DC = −5 A, (c) I DC = 5 A.

Figure 5 .
Figure 5. Changing the angle between the rotor discs.

Figure 5 .
Figure 5. Changing the angle between the rotor discs.

Figure 6 .
Figure 6.Distribution of magnetic field induction within the machine.

Figure 6 .
Figure 6.Distribution of magnetic field induction within the machine.

Figure 8 .
Figure 8.The relationship between the induced voltage and the angle .

Figure 7 .
Figure 7.The computational mesh generated by the FEM method.

Figure 7 .
Figure 7.The computational mesh generated by the FEM method.

Figure 8 .
Figure 8.The relationship between the induced voltage and the angle .

Figure 8 .
Figure 8.The relationship between the induced voltage and the angle α.

Figure 10 .
Figure 10.The relationship between the induced voltage and the angle  for basic and inverted stator connections.

Figure 10 .
Figure 10.The relationship between the induced voltage and the angle α for basic and inverted stator connections.

Figure 11 .
Figure 11.The relationship between the maximum value of the cogging torque and the angle .

Figure 11 .
Figure 11.The relationship between the maximum value of the cogging torque and the angle α.

Figure 11 .
Figure 11.The relationship between the maximum value of the cogging torque and the angle .

Table 1 .
Main mechanical and geometrical parameters.

Table 2 .
Number of mesh nodes in the main parts of the machine.

Table 2 .
Number of mesh nodes in the main parts of the machine.The computational mesh generated by the FEM method.

Table 3 .
Values of torques and pulsations for the angle  = 0°.

Table 4 .
Values of torques and pulsations for the angle α = 4 • .