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Numerical tools appear to be essential for modeling and designing devices based on superconducting materials. In this article different simulation results are presented, using a computer code based on the finite element method adopted for the resolution of the electromagnetic equations, in the case of an axisymmetric twodimensional problem, with this code we study the variations of the different electromagnetic quantities. The second generation superconductor has been modeled as an interesting diamagnetic material as inductive pulse sources. The performance in magnetic field resistance, energy storage and thermal stability of the ribbon, known as YBCO, makes it possible to broaden its field of application. Two categories of machines have been proposed and analyzed, the first is classic and the second uses a superconducting ribbon. In addition, a comparative study between the two proposed models is carried out and the results are analyzed and discussed.
energy storage, torque, secondgeneration superconducting ribbons, inductor, PMSM
Research into the heritage of superconductors has been relaunched since this discovery, while industrial applications may be possible. The conventional materials used in the various fields of electrical engineering are generally designed from an adequate assembly of conventional conductors and ferromagnetic materials, the latter have reached their limits of use because of the limitations of their performance, limitations due to heating in resistive materials, and saturation fields in magnetic materials. Superconducting materials seem at first sight to be able to defend against all their harmful drawbacks, thanks to their remarkable properties (zero resistivity, high current density, intense fields, etc.) [1–4]. But things are not quite as simple as we assumed, because superconductors also have their minor drawbacks, in particular the very low critical temperature (around 4.2K°). It is this major drawback which has limited the use of superconductors to particular fields, especially in applications requiring very high magnetic fields produced only by superconducting coils [5–8]. When we consider, to store a certain amount of energy, to use S.M.E.S. (Superconducting Magnetic Energy Storage), we obviously strive to propose a superconducting winding structure with the best performance. The objective is generally set to minimize losses during charging/discharging of the system. It is possible to set other criteria: the improvement of all electromagnetic quantities.
The appearance of superconductors known as: at high critical temperature (HTC) has really approved a new interest in these materials, and encouraged research on the applications of this type of material, in particular for the storage of electrical energy and the production of strong fields. Among the applications of electrical engineering, the most promising seems to be superconducting winding because it brings a new solution to conventional techniques [9].
The conventional electrical machines use copper, iron and permanent magnet as composite material having very high efficiency and very good compactness. These machines are used optimally in order to provide the best possible performance [10–12]. However, today new embedded applications require a gain in compactness and additional performance which is impossible to obtain with technological limitations that we know on conventional machines [13]. The advantages brought by the use of superconductors can be shown from a theoretical point of view by comparing a conventional synchronous motor and a machine of the same topology using superconductors [14]. One of the objectives of research on superconducting materials is related to the development of highperformance superconducting machines. Low critical temperature and high critical temperature superconducting machines are the two generations of known superconducting machines [15–17].
According to their design, they are classified into two categories:
Everyone knows that the principle of an electric motor is based on the interaction of the radial magnetic induction (Br) created by the inductor and the tangential field (Ht) produced by the armature and supplied with alternating current, generally threephase. This interaction generates an electromagnetic torque which is valid if the magnetic field is invariant along the Z axis [18]. Permanent magnet electric motors currently have the best powertoweight ratio, also called powertopower. But this is still insufficient, that's why disruptive technologies are being studied, such as the use of superconducting materials.
Under certain conditions, a superconductor can be used to trap a magnetic field, but also as a field shielding device to guide the magnetic field lines. The material is then used, no longer to channel the magnetic field as one usually did with the ferromagnetic materials, but to deviate it is the diamagnetic effect screen effect to create a new structure of a flux barrier superconducting machine [19–23].
When superconducting materials are used for the armature and the inductor, then the machine is said to be totally superconducting. Partially superconductive machine refers to a structure where only the inductor is superconductive, and the armature is conventional (copper coils). Totally superconducting machines aim to achieve the highest power densities, because the currentcarrying capacity of superconductors makes possible a significant increase in the inducing field and the linear current density of the armature. The thickness of the air gap is fixed by the mechanical separation, so it is substantially the same as standard electrical machines. In addition, yields are also increased, due to the elimination of DC losses. However, AC losses in superconducting conductors are still a major problem for the development of these machines today. They are due to the AC currents flowing in the superconductor but also to the variable electromagnetic environment in a machine [24, 25].
Partially superconducting machines have the potential to increase power density over standard electrical machine technologies by reducing or eliminating the use of ferromagnetic materials. The air gap is thicker than a standard or totally superconducting machine, due to the thickness of the thermal insulation between the armature and the inductor and the mechanical air gap. The currents in the superconducting parts are continuous (inductor); moreover the AC magnetic environment can be easily reduced by electromagnetic screens [26, 27]. AC losses are therefore less problematic, which explains why the majority of machines built to date are partially superconducting.
Different machines have been designed and tested by the GREEN laboratory for example: [14, 15, 18]
The particularity of these inductors is the use of combined superconducting wires and massive superconductors. Two superconducting coils are placed on either side of the inductor in the axial direction to create a strong magnetic field and bulk superconductors are used to modulate the magnetic field in the air gap (diamagnetic property).
Other HTc superconducting synchronous machines are produced by several research teams, for example, but not limited to: [17, 28, 30]
The objective of this study is to implant a secondgeneration superconducting materials having the particularity of suffering only from low losses (especially in the first thermally activated fluxflow (TAFF) and fluxcreep states), when an electric current passes through. We therefore quickly see an advantage concerning their integration in synchronous machines. Two categories exist for YBCO: bulk [29] and thin film (ribbon). For the application of this work, we have chosen to use tapes that are more flexible, more resistant to mechanical stress and therefore easier to roll up than a solid material. The structure presented in this work helps to get a high torque thanks to increase the radial field produced by the inductor which is made from superconducting ribbons. Another phenomenon that was also treated in this study is energy storage. We all know that the classic methods of storing electrical energy, using for the most part an intermediate energy (electrochemical, hydraulic, inertial storage). Magnetic energy storage, or S.M.E.S, uses a shortcircuited superconducting coil to store energy in magnetic form. Due to the absence of resistance in the superconducting ribbon, this energy can be stored almost indefinitely.
The device constituting the studied inductor is a conductive type deposited with YBCO, it consists of three main layers (shunt, superconductor and substrate) and several buffer layers. These last layers have a major and important role in machine operation. They also make possible the creation of chemical diffusion barrier preventing the pollution of the superconductor by the elements of the substrate and preventing the oxidation of the latter during the YBCO deposition phase. The simplified architecture adopted for a ribbon of length L rub is described in figure.1.
Secondgeneration superconducting ribbons have been developed with the aim of getting as close as possible to the native properties of superconductors perfectly textured by epitaxial growth [28, 31]. The architecture of these ribbons is made in the form of a sandwich of three thin layers: the substrate, the superconducting layer and the shunt, (Fig. 1).
Figure 1. The description of the ribbon model
Each layer has a role in the operation of the winding as developed in the proposed model. The substrate acts as a support, its main function is to dissipate the excessive heat produced during operation and thus protect the superconducting element. The thickness of the substrate is around 50 μm, the alloys used are Nickelbased (NiCr, Inconel, Hastelloy and Constantan).
They make possible the adaptation of mechanical stresses, resulting from the difference in coefficient of thermal expansion between the superconductor and the substrate, and the difference in lattice parameter between these two materials.
The superconducting layer has a thickness of 1 μm and the shunt has an average thickness of 20 μm, it acts as a thermal and electrical stabilizer. The material used for this layer is silver [32–37].
3.1 Putting into equations of the PMSM in the frame (abc)
$[V]=\left[R_S\right][i]+\frac{d[\varphi]}{d t}$ (1)
$[V]=\left[V_a V_b V_c\right]^T$: Vector stator voltages.
$[i]=\left[i_a i_b i_c\right]^T$: Stator current vector.
$[\varphi]=\left[\varphi_a \varphi_b \varphi_c\right]^T$: Stator flux vector.
$\left[R_s\right]=\left[\begin{array}{ccc}r_s & 0 & 0 \\ 0 & r_s & 0 \\ 0 & 0 & r_s\end{array}\right]$: Matrix of the equivalent resistance of a stator winding.
The matrix of the inductances, which establishes the linear relations between the fluxes and the currents, is the following:
$[\varphi]=\left[\mathrm{L}_{\mathrm{s}}\right]\left[\mathrm{i}_{\mathrm{s}}\right]+\left[\varphi_{\mathrm{f}}\right]$ (2)
$\left[L_s\right]=\left[\begin{array}{ccc}I_s & M_s & M_s \\ M_s & I_s & M_s \\ M_s & M_s & I_s\end{array}\right]$: Proper inductance matrix of a stator phase.
$\left[\varphi_f\right]=\left[\varphi_{a f} \varphi_{b f} \varphi_{c f}\right]^T$: Flux vector created by the magnet through the stator winding.
The mechanical equation is written as follows:
$\mathrm{J} \frac{\mathrm{d} \Omega}{\mathrm{dt}}=\mathrm{C}_{\mathrm{em}}\mathrm{C}_{\mathrm{r}}\mathrm{C}_{\mathrm{f}}$ (3)
$C_{\mathrm{f}}=\mathrm{f}_{\mathrm{c}} \Omega$ (4)
$\Omega=\frac{\omega_r}{p}$ (5)
With:
C_{r}: Resistant torque.
C_{em}: Electromagnetic torque.
C_{f}: Friction torque.
J: Moment of inertia.
f_{c}: Coefficient of friction.
p: Number of pole pairs
$\omega_r$: Electric rotor speed
3.2 Park transformation
In order to remove the nonlinearity of the system of differential equations, changes of variables is required to reduce the complexity of this system. In threephase electrical machines, this change of variable consists in transforming the three windings relating to the three phases to orthogonal windings (d, q), rotating at a speed ω. With this transformation, a representation in the threephase coordinate system (a,b,c) is transformed to a representation in a Cartesian coordinate system of axes (d,q).
From this transformation, the equations are written as follows:
$V_d=R_s i_d+\frac{d \varphi_d}{d t}\omega_r \varphi_q$
$V_q=R_s i_q+\frac{d \varphi_q}{d t}+\omega_r \varphi_d$ (6)
$\varphi_d=L_d i_d+\varphi_f$
$\varphi_q=L_q i_q$ (7)
By replacing the expressions of the fluxes $\varphi_d$ and $\varphi_q$ in the system
$V_d=R_s i_d+L_d \frac{d i_d}{d t}\omega_r L_q i_q$
$V_q=R_s i_q+L_q \frac{d i_q}{d t}+\omega_r L_d i_d+\omega_r \varphi_f$ (8)
The state form is obtained:
$\frac{d i_d}{d t}=\frac{1}{L_d}\left(V_dR_s+\omega_r L_q i_q\right)$
$\frac{d i_q}{d t}=\frac{1}{L_q}\left(V_qR_s i_q\omega_r L_d i_d\omega_r \varphi_f\right)$ (9)
$\left[\begin{array}{l}\dot{I}_{\mathrm{d}} \\ \dot{\mathrm{I}}_{\mathrm{q}}\end{array}\right]=\left[\begin{array}{cc}\frac{R_s}{L_d} & 0 \\ 0 & \frac{R_s}{L_q}\end{array}\right]\left[\begin{array}{l}\mathrm{I}_{\mathrm{d}} \\ \mathrm{I}_{\mathrm{q}}\end{array}\right]+\omega_r\left[\begin{array}{cc}0 & \frac{L_q}{L_d} \\ \frac{L_d}{L_q} & 0\end{array}\right]\left[\begin{array}{c}\mathrm{I}_{\mathrm{d}} \\ \mathrm{I}_{\mathrm{q}}\end{array}\right]$
$+\omega_r\left[\begin{array}{c}0 \\ \frac{\varphi_f}{L_q}\end{array}\right]+\left[\begin{array}{cc}\frac{1}{L_d} & 0 \\ 0 & \frac{1}{L_q}\end{array}\right]\left[\begin{array}{l}\mathrm{V}_{\mathrm{d}} \\ \mathrm{V}_{\mathrm{q}}\end{array}\right]$ (10)
The electromagnetic torque is expressed by the partial derivative of the electromagnetic energy with respect to the geometric angle of rotor rotation expressed below:
$C_{e m}=\frac{d W_e}{d \theta_{g e o}}=P \frac{d W_e}{d \theta}$ (11)
According to Park, the expression for the transmitted power is as follows:
$P(t)=\frac{3}{2}\left(V_d i_d+V_q i_q\right)$ (12)
By replacing V_{d}V_{q} by their expressions we will have:
$P(t)=\frac{3}{2}\left[R_s\left(i_d^2+i_q^2\right)+\left(\frac{d \varphi_d}{d t} i_d+\frac{d \varphi_q}{d t} i_q\right)+\frac{d \theta}{d t}\left(\varphi_d i_q\varphi_q i_d\right)\right]$ (13)
$\frac{3}{2} R_s\left(i_d^2+i_q^2\right)$: Represents the power dissipated in Joules losses in the stator windings
$\left(\frac{d \varphi_d}{d t} i_d+\frac{d \varphi_q}{d t} i_q\right)$: Represents the change in magnetic energy stored in the stator windings
$\left(\varphi_d i_q\varphi_q i_d\right)$: Represents the electromagnetic power.
Knowing that:
$P_e=C_{e m} \omega_r$ (14)
He comes:
$C_{e m}=\frac{3}{2} P\left(\varphi_d i_q\varphi_q i_d\right)$ (15)
The expression of the electromagnetic torque as a function of the currents is as follows:
$C_{e m}=\frac{3}{2} P\left[\left(L_dL_q\right) i_q i_d+i_q \varphi_f\right]$ (16)
The study of the dynamic behavior of a synchronous electric machine requires a transient magnetic analysis with electromechanical coupling, where the mechanical equation characterizing the movement of the rotor is added to the magnetic or electromagnetic model of the machine.
3.3 Maxwell equations
Macroscopically, Maxwell's equations remain valid to describe the electromagnetic phenomena that occur in a superconductor, they are written.
$\overrightarrow{\operatorname{Rot}} \overrightarrow{E}=\frac{\partial \overrightarrow{\mathrm{B}}}{\partial \mathrm{t}}$ (17)
$\operatorname{Div} \overrightarrow{\mathrm{B}}=0$ (18)
$\overrightarrow{\mathrm{B}}=\overrightarrow{Rot} \overrightarrow{A}$ (19)
$\overrightarrow{Rot} \overrightarrow{H}=\overrightarrow{\mathrm{J}_{\mathrm{t}}}$ (20)
For a magnetic medium:
$\vec{B}=\mu_0 \vec{H}$ or $\vec{H}=v \vec{B}$ (21)
For a dielectric medium:
$\vec{D}=\varepsilon \vec{E}$ (22)
For a conductive medium:
$\vec{\jmath}=\sigma \vec{E}$ (23)
For a superconducting medium:
In a superconducting medium the relationship between electric field and electric current density is nonlinear, i.e.
$\overrightarrow{J_c}=\sigma(\vec{E}, \mathrm{~T})$ (24)
The development of the MAXWELL equations governing the electromagnetic behavior of the studied domain leads to the following equations [4,5],where the electromagnetic and thermal coupling is provided by the alternating coupling mode. To model the electromagnetic behavior of the presented study, we have adopted the formulation in magnetic vector potentials A and in electric scalar potential V; this is described by the equation presented below.
$\nabla \times(v \nabla \times \mathrm{A})\nabla(v \nabla \cdot \mathrm{A})+\sigma(E, T)\left(\frac{\partial \mathrm{A}}{\partial \mathrm{t}}+\nabla V\right)=\mathrm{J}_s$
$\nabla \cdot\left\{\sigma(E, T)\left(\frac{\partial \mathrm{A}}{\partial \mathrm{t}}+\nabla V\right)\right\}=0$ (25)
$v$ and $\sigma$ represent respectively the magnetic reluctivity and the electrical conductivity of the superconductor. Concerning the apparent electrical conductivity of the superconducting material, in its nondissipative state, it is defined by the ratio of J on E; this ratio is deduced from the characteristic EJ of the superconductor given by the following relation.
$\sigma_S(E, T)=\frac{J}{E}=\frac{J c(T)}{E c}\left(\frac{E}{E c}\right)^{\frac{1}{n(T)}1}$ (26)
With:$J c(T)=J_0 \frac{\left(1\frac{T}{T c}\right)}{\left(1\frac{T_0}{T c}\right)}$ (27)
This relation reflects the behavior of the superconductor in a nondissipative state, to complete the expression of the electrical conductivity of the superconductor in the dissipative regime. An additional term $\sigma_n$ is added translating the increase in the resistance of the superconductor. Thus, the apparent electrical conductivity of the superconductor is deduced by the relation.
$\sigma(E, T)=\sigma_s(E, T)+\sigma_n(T)$ (28)
where, J_{C} and E_{C} respectively represent the density of the critical current and the critical electric field. According to relation (19), the apparent conductivity of the superconductor depends on the electric field E and the temperature T reached within the material. The electric field E will be determined from the resolution of the electromagnetic problem described by the partial differential equation presented by the formulation (17). The temperature will be determined from the resolution of the heat diffusion problem presented by
$\rho C_p(T) \frac{\partial T}{\partial t}\nabla \cdot(\kappa(T) \nabla T)=W$ (29)
where, λ(T), ρ, Cp(T) are respectively the thermal conductivity in (W / K / m), the density in (Kg / m^{3}) and the specific heat of the material in (J / K / Kg), W is a power density in (W / m^{3}), it expresses all the losses generated in the coated superconducting materials given by the following equation:
$W=E . J$ (30)
We will limit ourselves in this section to a purely “magnetic”study relating to objectives relating to the distribution of the energy stored in the inductor. Compared to other types of storage, magnetic energy storage does not require reconverting the energy, directly accessible in electrical form, and allows flexible and almost infinite cyclability. So it has very high storage efficiency. The magnetic energy stored in the system is then obtained as the halfproduct of the equivalent total inductance of the system times the square of the current flowing in each strip.
Energy and power are expressed as in equations (31) and (32)
$E=1 / 2 L I^2=1 / 2 \iiint B H d x d y d z$ (31)
$P=d E / d t$ (32)
where, ‘I’ is the current through the coil and ‘L’ is the superconducting coil inductance, the SMES stores the energy as circulating current, and provide the energy with instantaneous response.
This part sets up the choices made on the materials and properties that will be used for the simulations that will follow. The various parameters used in the simulation are presented in the following three tables:
Table 1. Simulation parameters for the superconducting material YBCO and Copper
Symbol 
Amount 
Value (YBCO) 
Value (Copper) 
γ 
Volumic mass 
5.4 g/cm^{3} 
8,96 g/cm^{3} 
C_{p} 
Specific heat 
150 J/(Kg.K) 
385 J/(Kg.K) 
λ 
Thermal conductivity 
5 W/(m.K) 
380 W/(m.K) 
h 
Cryogenic fluid convection coefficient 
400 W/(m^{2}.K) 
 
Tc 
Critical temperature 
93 K 
 
Ec 
Critical electric field 
1 µV/cm 
 
J_{c0} 
Critical current density at 77 K under zero field 
500 A/mm^{2} 
 
n_{0} 
Exponent n at 77 K under zero field 
20 
 
Table 2. Parameters of the studied machine
Amount 
Value 
Resistance of a winding 
R=1.4 Ω 
direct inductance 
Ld=6.6 mH 
quadratic inductance 
Lq=5.8 mH 
Number of poles 
P=8 
Friction 
F=0.0003881 
Number of teeth/pole 
6 
Magnet 
NdFeB 
Lamination type 
M27035A 
Remanent flux density (NdFeB) 
1.2 T 
Energy density (NdFeB) 
380 (KJ/m^{3}) 
coercive field (NdFeB) 
860 10^{3} (A/m) 
Table 3. Simulation parameters for the materials that make up the ribbon
Symbol 
Amount 
Value 

Silver 
Hastelloy C276 

γ 
Volumic mass 
10500 Kg/m^{3} 
8890 Kg/m^{3} 
C_{p} 
Specific heat 
235 J/(Kg.K) 
427 J/(Kg.K) 
λ 
Thermal conductivity 
429 W/(m.K) 
7 W/(m.K) 
σ 
Electrical conductivity 
61,6 ×10^{6} S/m 
77 ×10^{4} S/m 
In this phase, we will focus on the study of the temporary evolution of different electromagnetic quantities within two proposed machines. The proposed inductor is composed from two coaxial superconducting ribbon solenoids and a screen. The two solenoids are fed by a direct current having two different directions, and generate a magnetic field as shown in Figure 2.
Figure 2. Structure of the studied inductor
Table 4. Geometric parameters of the studied inductor
Amount 
Value 
Thickness of solid YBCO 
15 mm 
Width of solid YBCO 
40 mm 
Length of the solenoid 
60 mm 
Solenoids outer radius 
65 mm 
Solenoids inner radius 
54 mm 
This section is devoted to the different simulation results obtained, the study is focused on the variations of the different electromagnetic parameters, namely (current, speed, magnetic flux density, electromagnetic torque, etc.) of the two machines (conventional and superconducting).Our code, based on the finite element method adopted for the resolution of the equation describing the behavior of the ribbon implanted in the inductor of the machine in the case of an axisymmetric twodimensional problem.A comparative study between the two proposed models is conducted.
Figure 3. Variation of current Id as a function of time
Figure 4. Variation of current Iq as a function of time
Figure 5. Variation of speed W as a function of time
Figure 6. Torque variation Ce as a function of time
The figures above (36) represent the development of some fundamental variables of two studied machines, namely the direct and quadrature components of the current and the speed. In noload operation, in the Permanent Magnet Synchronous Machine (PMSM), there is an excessive inrush of current when the motor is powered up in transient state which stabilizes to give rise to a sinusoidal shape of constant amplitude.
There is also an increase in the amplitude of stator currents.According to the electromagnetic torque equation, its shape is derived from the shape of the current in quadrature Iq, it is found that the effect of the salience slightly influences the main torque.
From these results, the speed oscillates in the transient regime until it stabilizes in the steady state at a fixed value. The speed up time is determined by the total inertia around the rotating shaft, the motor not being loaded, the speed reached is equal to the synchronism speed.
Now, regarding the superconducting machine, from the presented results where a comparison has been conducted in terms of current (Id, Iq), rotational speed (ω) andthe torque (Ce),we clearly see a significant difference between the results of the characteristics of two machines in particular the reduction of transient oscillations and these quantities stabilize more quickly before even the end of the transient regime. It is also noted that these simulation results have shown the interest and the efficiency of the superconducting materials introduced in the design of an electric machine. These significantly improve thermal stresses during the operating process by reducing joule losses. Thus they can considerably extend the life of superconducting materials.
Fig. 7 presents and compares important characteristics of these two inductors, it is observed that the position of the rotor starts from a low value and increases as a function of time. It is also observed that the superconducting inductor machine has a superior position compared to the conventional machine.
Figure 7. Rotor position as a function of time
In figure 8, the rotational speed is presented as a function of time for the two studied cases. From these simulation results, it can be observed that the curves follow the same trajectory. Nevertheless, the speed values obtained by the superconducting inductor are much higher than those obtained in the case of a conventional inductor. It can be noted that this speed reaches a maximum value of 6950.54deg/s for a conventional machine and a value of 9998.84deg/s for a superconducting machine.
Figure 8. Rotation speed (rotor)as a function of time
Figure 9. Stored energy as a function of time
Figure 9 shows the stored energy as a function of time, it is known that in order to let the current flow in the inductor, it is necessary to supply energy to this conductor. This energy subsequently is stored by the magnetic field of the inductor and recoverable once the current is switched off.
Therefore starting from the expression relating energy and power, it appears according to the results presented in figure 9, that when we consider a current passing through the superconducting inductor this stored energy reaches its maximum at 0.94 J, meanwhile in a copper inductor inductance at 0.62 J. This is explained by the fact of the important characteristics. Indeed the storage and conservation of electrical energy poses serious problems, as for electrochemical methods, their inability to quickly exchange energy with external circuits. The use of conventional electromagnetic devices for the storage of electrical energy hardly leads to better performance even with the use of good conductors, but unlike conventional storage, electromagnetic storage by secondgeneration superconducting coils allows energy exchanges. fast energy with electricity use network. brought by the introduction of superconducting materials which behave like perfect conductors in a diamagnetic medium. The calculation code gives the inductance matrix of the simulated coil system. The inductances obtained are those of the simulated coil systems but where the coils are considered to be made of a single conductor. Each coil being in fact made up of thousands of turns (their number is obtained by dividing the total section of the winding by the section S of the superconducting tape), it is advisable to weight each term of the inductance matrix by the square of the number of turns for the proper inductances and by the product of the numbers of turns of the coils considered for the mutual inductances.
In this paper, an inductor design strategy to improve the electromagnetic properties in the permanent magnet synchronous machine using secondgeneration superconducting material was proposed. The efficiency of the proposed model was tested using our calculation code under MATLAB.
The results obtained for the two models are compared for a time step of 0.09 s. This choice can be explained by the large number of mesh elements presented in the field discretized with the FEM. This would significantly increase the computation time. Superconducting machines are known to represent an encouraging means of making superconductivity commercially operational in real power systems. The rapid development of type II superconducting materials is linked to the improvement of their performance.
The results obtained showed that the objectives of the present contribution are achieved and indeed, the electromagnetic characteristics such as torque, rotational speed and stored energy are improved both in steady state and in transient state. These results demonstrate that the use of thin films has considerably improved the electromagnetic behavior of the superconducting inductor by the superconducting coils which must be supported by materials with excellent mechanical properties and poor thermal properties. The study presented in this work made possible the illustration of the superconducting inductor principle towards the optimization on cycle of the synchronous machines with magnets by an original approach entirely digital.
The future trends will be mainly dedicated to the sizing and optimization of the superconducting inductor taking into account the magnetic losses that may occur during machine operation. It is a question of establishing a model of magnetic losses for the latter.
The authors would like to thank Editors and reviewers for their pertinent comments and suggestions to improve our work. The authors thank the General Directorate of Scientific Research and Technological Development (DGRSDT).
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