Characterization of linear motor flux pump using H-formulation

H-formulation is an effective method for electromagnetic simulation analysis of superconductors. This paper innovatively uses H-formulation to simulate and analyze the operation characteristics of magnetic flux pump operating in linear-motor pattern. Via COMSOL Multiphysics, this paper establishes a two-dimensional electrical equipment simulation model that comprehensively compares excitation conditions of triangular and trapezoidal current waves to investigate the effects of the working excitation waveforms’ frequency and amplitude on the operating features of the equipment. The following operation control laws are revealed: the current amplitude has a linear control effect on the mean value of superconductor surface electric field and the average amount of air-slot density of magnetic flux, the current frequency has no effect on the air-gap magnetic field waveform, and the regulation effect of the excitation frequency on the average value of superconductor surface strength of electric field is related to the waveforms. Finally, the equivalent circuit of the pumping model is established, and two interrelated L-R equivalent circuits are designed by quadratic substitution, and the final pumping current reaches 43A.


Introduction
High-temperature superconducting magnets offer numerous advantages [1] , including high currentcarrying density and magnetic field stability [2] .They are utilized in various fields such as electric power, energy, medical care, and aerospace [3] .They are employed in large-scale offshore wind power equipment, as well as in large-scale scientific instruments.High-temperature superconducting magnets face challenges such as joint resistance, AC loss, and flux creep, preventing them from operating in a constant current closed-loop state [4] .However, the linear motor flux pump serves as an innovative wireless energy transmission equipment that offers advantages like lightweight design, low loss, and minimal electromagnetic noise [5] .The magnetic flux pump operating in linear pattern is a new technical support that uses pumping flux to ensure the stable operation of high-temperature superconducting magnets [6] .However, there is currently limited simulation analysis of this pump using the finite element method.Therefore, conducting further finite element simulation analysis is important to establish a reliable reference basis for its operation.
The finite element (method) were applied in simulating the electromagnetic processes of superconductors with nonlinear resistivity characteristics [7] , and H-formulation has been successfully applied to simulate the electromagnetic behaviors of high-temperature superconductors because of its wide range of applications, direct solution variables, and easy-to-impose boundary conditions [8] .COMSOL Multiphysics is a commercial multi-physics software that provides technical support for finite element simulation calculations using H-formulation [9] .
In this paper, a reliable and efficient two-dimensional FET model using H-formulation is presented for analyzing linear motor-type flux pumps, which is the closest planar model.In order to couple the external current excitation with the magnetic property, Ampere loop with displacement current ignored is used to simulate the external excitation.We analyzed the impact of two different current excitation waveforms, including changes in amplitude and frequency, on the output electromagnetic parameters of the two-dimensional FET simulation model.The purpose of this study is to establish an effective control law for the magnetic flux pump working in straightness motor form and offer assistance for its operation.Finally, a pumping process equivalent circuit model was simulated and analyzed using an integrated finite element analysis model based on MATLAB-Simulink to obtain steady-state current data.we use two-dimensional FET model to analyze straightness-motor magnetic flux pump and establish an equivalent electric circuit.This provides strong support for the development of high-temperature superconducting flux pump technology and superconducting magnet technology.

Simulation FET model
The H-formulation method treats the strength of magnetic field H as the crucial element to solve directly, and the associated system of solenoid equations combined with the resistivity characteristics in different materials to solve the double-spin equation for the global vector field.The H-formulation method adopts the core themes of finite element to deal with the classical equation of electromagnetic induction Proposed by Faraday [10] .
Solve the equation for the spin degree considering only the conduction current, ignoring displacement currents.
  = HJ (2) Superconducting regions characterized by nonlinear resistivity and homogenization.In Equation (3) E0 and Jc is the critical parameters for loss of superconductivity, and n is the superconducting resistivity modeling tuning parameters.
For the non-conducting area, we can install its solving resistivity to 1 Ω•m to ensure the calculation accuracy and accelerate the convergence speed.
This simulation is assumed to be an ideal heat transfer model, regardless of the temperature on Jc, The applied current for the simulation is small, while the air gap permeability is low, so ignore the effect of the magnetic property intensity on the Jc, so it is assumed that the Jc = 10 8 A/m 2 .E0 is usually taken as 10 -4 V/m.n is the exponential factor, and generally goes to 25-31.
Equations ( 2) and ( 3) are combined with the permeability and resistivity of each material to obtain the final double-spin equation.
For the 2D model, the magnetic field strength H can be expressed as the following vector: H= [Hx,Hy], and the density ruggedness of current J and the electric field strength E have only z-directions perpendicular to the plane where the magnetic field strength is located, so we can get.

2D geometric model
The 2D geometry of this simulation model is shown in Figure 1.The 2D geometry of this linear motor flux pump consists of air domains, superconducting strips, current windings, and a stator core.This 2D geometry is the axial section of the 3D structure of the pumping equipment in open looping operation.

Pumping-process equivalent circuit model
The straightness-motor type magnetic flux pump is connected to the superconductor coil load through the center superconductor, which is equivalent to a direct current source, and the superconductor coil load is equivalent to an R-L load, of which the linear motor flux pump pumping process equivalent circuit diagram is indicated in the Figure 2. 4 pairs of energized solenoids are excited to arise pumping voltage by energizing an excitation current waveform with an angle difference of 90° to produce a traveling wave magnetic field [11] .The second iteration computational circuit simulation circuit model is built based on MATLAB-Simulink as Figure 3: Simulation modeling of linear motor flux pumping circuits using numerical simulation, using superconducting resistor time-domain averages as the primary computational current and as the secondary computational signal for secondary computation.
The equivalent DC supply voltage and the equivalent resistance component of the superconducting load are calculated as follows: (1 Equation ( 7) defines the scale factor in k, while αf represents the eddy current loss.The value of α is typically very small, and in order to keep the value of loss low, the device is typically operated at 10-20Hz.Equation ( 8) describes R0 and Rdyn as constants in this model, while Equation ( 9) is used to describe the dynamic characteristics of superconducting resistors.

Analysis of air-gap magnetic flux density results
we obtain the results in figure 4 using triangular and trapezoidal waveforms as excitation sources via changing the waveform, amplitude, and frequency of the excitation source.It can be observed in the Figure 4: when the excitation source amplitude and frequency are kept constant, the amplitude of air-slot magnetic throughput density via trapezoidal wave excitation is slightly larger than same output parameter via triangular wave excitation.keeping the excitation source amplitude at 5A and adjusting the frequency of the excitation source, we can discover that the air-gap waveform remains constant, so it is indicated by adding * to 5A5Hz in figure 4.
For the same excitation waveform, the amplitude of density of magnetic throughput in air-gap and the amplitudes of excitation sources are positively correlated.Keeping the excitation frequency at 5Hz calculate the average value of the air-slot magnetic throughput density waveforms under diverse excitation amplitudes, to obtain the Table 1. Figure 5 explicitly indicates the results of the fitting operation that the excitation source amplitude of the air-gap average magnetic flux density presents a linear control effect.

Analysis of surface electric field strength results
The electric intensity on the membrane of the central superconductor is averaged using the same method as that used in 3.1 for calculating the value of the mean density of magnetic throughput at the air slot.
The mean intensity at the surface of the superconductor under the action of triangular and trapezoidal wave excitation currents is investigated in relation to the excitation current amplitude and frequency according to the single-variable principle.Keeping the frequency of the excitation current constant f=5Hz, change the current amplitude to investigate the variation of the surface average electric intensity of the superconductor with the amplitudes of the excitation sources, as shown in the Table 2, and according to the data in the Table 2 plot the Figure 6.Keep the excitation current amplitude constant Am = 5A, change the current frequency to investigate the superconductor surface average electric field strength with the excitation source amplitude changes, the data as in Table 3, according to the data in the table to plot the Figure 7.The simulation calculation model explicitly indicates that the control effect of the excitation source amplitude on the mean calculation results of the intensity of electric field at the surface of the superconductor is linear, while the control effect of the excitation source frequency on the mean number of the electric field strength on the surface of the superconductor is related to the waveform, and the excitation effect and control characteristics of trapezoidal wave are better than those of triangular wave.For the triangular wave type current source, the control of frequency on the average value of the final electric intensity is nonlinear, and the average calculation results of the electric field strength increases faster and then slower with the increase of frequency, while for the trapezoidal wave type current source, the control of frequency on the final electric intensity is linear in tested frequency range.The reason for the different frequency control effects of the two waveforms is presumed to be Faraday's law, for the triangular waveform current source, when the frequency is lower, the value of the reaction potential generated by the obstruction of the magnetic field alternating with time is lower, so the final average value of the electric field strength increases faster with frequency at lower frequencies, while when the frequency is higher, the value of the reaction potential of the obstruction of the effect is larger, inhibiting the mean calculation number of the electric intensity to increase, so The final mean result of the electric intensity increases more slowly with frequency at higher frequencies.For the trapezoidal waveform current source, because of its waveform characteristics of the time change with the current output unchanged part, so for the reaction potential in the high-frequency part of the increase in the inhibition, so that the calculated mean number of the electric intensity in the measured frequency corridor is a linear change.When the frequency is gradually increased, the shape of the trapezoidal wave is close to the triangular wave, the part of the current output that changes with time and remains unchanged decreases, and the value of the counter electromotive force increases, thus appearing similar to the triangular wave-type current source in the frequency of the mean calculation number of the electric intensity of the nonlinear control effect.

Analysis of pumping-process equivalent circuit results
The numerical simulation of the actual pumping process is carried out according to the simulation electrical result model in Figure 3, and the simulation parametric index are adopted in Table 4.The calculation and analyzation data are displayed in Figure 8.
The initial calculation selects a fixed resistance parameter as the average value of the superconducting resistor Rs, and sets the values of the parameters of the completed joint resistance R0, the dynamic resistance Rdyn, and the equivalent inductance L, to obtain the initial calculation of the pumping current I1.Then, the variable resistance model of the superconducting resistance Rs is constructed in the form of the expression of the controlled voltage source.The pumping current I1 obtained from the initial calculation is calculated by the superconducting resistance function operation module as the control signal of the controlled voltage source to complete the modeling process of the superconducting resistor Rs.At the same time, the other parameters are kept the same as those of the initial calculation, and the iterative calculation is completed to obtain the final value of the pumping current I2.The simulation results show that the superconducting resistor enhances the complexity of the system, upgrading it from the initial first-order system to a second-order system with a certain amount of overshooting.The final pumping current can reach 43 A. The simulation results are compensated for an initial current of 0A.The results indicate that the flux pumping device can effectively compensate for the load, and the pumping efficiency can be further rationalized by extending the extent l0 of the superconducting strip and the effective coupling area with the magnetic poles.

Conclusion
This paper establishes a 2D finite element model of the magnetic flux pump working in straightness motor form.Under the excitation of different current waveforms, the changes of the output electromagnetic parameters of the simulation FET model are explored, and the working characteristics of the linear motor flux pump are completely analyzed and reasonably explained.The following control laws are revealed: the excitation source amplitude has a linear control effect on the density of magnetic flux field in the air slot, while the excitation source frequency has no effect on the travelling waveform in the air slot; the excitation source amplitude has a linear control effect on the electric intensity on the surface layer of the superconductor, and the frequency of the excitation source appears to be saturated because of the electromagnetic induction.After completing the finite element simulation analysis of the device, the equivalent circuit of pumping model in close-loop working mode is established, and the simulation platform of the circuit is built to simulate the pumping process of the linear motor device, and the final pumping current reaches 43A. 0

Figure 1 .
Figure 1.The 2D geometry of the axial section analyzation model.

Figure 2 .
Figure 2. Equivalent circuit of pumping process excited by traveling wave.

Figure 4 .
Figure 4. air-gap Waveforms for different excitation amplitudes and frequencies, where triangular wave is denoted by 'Tri' and trapezoidal wave is denoted by 'Tra'.

Figure 5 .
Figure 5. Linear control results of triangular and trapezoidal wave excitation current amplitude (Am) on the value of air-gap average magnetic flux density (Bav).

Figure 6 .
Figure 6.Linear control results of triangular and trapezoidal wave excitation current amplitude (Am) on the value of surface average electric field strength (Eav).

Figure 7 .
Figure 7. Linear control results of triangular and trapezoidal wave excitation current frequency (f) on the calculation number of surface average electric intensity (Eav).

Table 1 .
The calculated averages for the density of the magnetic throughput at different excitation source amplitudes (f=5Hz).

Table 3 .
The surface average electric field strength at different excitation source frequency (Am=5A).