What is MD in Fig. 1?
MD is the abbreviation of 'magnetic dam'.This explanation would be added in the describtion of Fig.1
Show the explanation of all the symbols in all the equations.
Explanations would be added after Eq.1, Eq.2, Eq.3 and Eq.10.
P. 2 left L. 45: To ignore the copper resistance is inappropriate. The current induced in the copper depends on its resistance, and part of energy of LTS magnet is consumed with the copper resistance. So the copper resistance plays an important role.
Ignoring the copper resistance is only intended to facilitate the understanding of the circuit structure, indicating the generation of inductive currents and inductive potentials.
This does not mean that the resistance of magnetic dam is not taken into account throughout the analysis.
This is completely a misery of my statement, that caused unnecessary misunderstanding. The wording has been modified in the text.
Equation (4) is not the equivalent radial resistance.
I made a mistake when I combined the change of superconducting resistance with the Ohm's law:
$$U = {10^{ - 5}} \cdot l \cdot {I^n}/I{c^n} = {R} \cdot I$$
$${R} = {10^{-5}} \cdot l \cdot {I^{(n - 1)}}/I{c^n}$$
where ${10^{-5}}$ is the quench criterion, ${l}$ is the line length, ${I_c}$ is the critical current, and $n$ is the $n$-value of REBCO tapes.
How did you calculate all the self and mutual inductances. As the inductance values are very important, please show the value of self and mutual inductances of HTS1, HTS2, magnetic dam, and LTS.
The method of mutual inductance calculation is vector magnetic potential integration.
Regarding one turn of superconducting tape as an ideal current loop $l_1$, the flux linkage between it and another loop $l_2$ can be obtained by the following integral formula:
$$\Psi {\rm{ = }}\oint_{{l_2}} {\vec A \cdot d\vec {l_2}} $$
Where A is the vector magnetic potential of the current element in l_1 in l_2:
$$\vec A = {{{\mu 0}} \over {4\pi }}\oint{{l_1}} {{I \over R}} \cdot d\vec {l_1}$$
Where R is the distance between the two current sources
So the mutual inductance of the two current paths is:
$$M = {{{\mu 0}} \over {4\pi }}\int{{l_1}} {\int_{{l_2}} {{{d\vec{l_1} \cdot d\vec{l_2}} \over R}} } $$
In this way, the mutual inductance of various coils can be obtained by means of vector magnetic potential integration
The mutual inductance between a ring and a cylinder is similar. The vector magnetic potential of the cylinder position is integrated according to the cross-section of the cylinder, and the mutual inductance can be obtained by combining the current density distribution.
Since each turn of the wire must be calculated for mutual inductance with the magnetic dam, in order to reduce the calculation time, I divided the cylindrical section into a grid of 1mm*1mm, and calculated the mutual inductance with this rectangular cross-section ring as the object. Calculate instead of points.
The finally calculated mutual inductance matrix is in the appendix parameter_cal_mag_dam.mat. The file also contains other parameters, such as radial resistance, mutual inductance between superconducting parts, etc.
What temperature HTS/LTS magnets drive at? Is the temperature of the magnetic dam the same?
Give us the specifications of REBCO tape, such as Ic, n-value, tape width, thickness, …. And need the key sepc. of magnets: number of turns, the contact resistivity/resistance, …. Also need the resistance and inductance of magnetic dam.
The whole magnet was set in liquid Helium.
The temperature of the magnetic dam in simulation was set at 4.2K.
The parameters of the magnet can be referred to:
World record 32.35tesla direct-current magnetic field generated with anall-superconducting magnet. Superconductor Scienceand Technology 33(3), 03LT01 (2020).doi: 10.1088/1361-6668/ab714e.
The parameter table is available in Parameters of the REBCO insert.pdf
What software was used for stress simulation?
We used the magnetic field (mf) and solid mechanics(solid) Physics of COMSOL to simulation the stress of the magnetic dam.
Fig. 5: Why are the current shown by 100 ms? Do the currents decrease after 100 ms?
Fig. 5: What does "0.5" in the y axis mean?
Fig. 5: How much are Ic and the initial current value? Does not the current exceed Ic?
The current change within 100 ms shows a monotonous current change as a whole and, at the same time, demonstrate the protective effect of HTS-2 on HTS-1.
Simulation shows that, the critical current of the double cakes is approximately 250A in the parts with the worst magnetic field conditions (located at both ends, B = 3.5 Tesla, T = 4.2 Kelvin). The critical current with little vertical magnetic field is above 500A(B = 1 Tesla, T = 4.2 Kelvin ).
Fig. 6: "modulus"?
Fig. 6: Why are these maps at 0.24 s show? 0.24 s is special?
Fig. 6: Please show the magnetic flux density map before quench of LTS. The comparison is a good way to understand.
'Magnetic flux density modulus' is a numerical name used in our simulation process. I forgot to convert it into a more understandable style in the article. Among non-magnetic materials, Magnetic flux density modulus is the size of the magnetic field. This mistake has been corrected in the revision.
The deformation of the magnetic dam reaches the maximum at 0.24s.
At this time, both HTS-1, HTS-2 and the magnetic dam have relatively high current density, which can show the effective state of the magnetic dam.
Figs. 7 & 8: The maps of what time are shown? At that time, is the deformation maximum?
As stated in the picture description, these two pictures showed the maximum deformation, the time points are 0.24 seconds and 0.21 seconds respectively.
The stainless steel cylinder was allocated at outside of copper cylinder. The thickness of copper is constant?
As mentioned in the article, the total space for the magnetic dam is limited. The thickness of copper will be smaller with a stainless steel shell outside.
When charing magnet, how much current is induced in the magnetic dam? How about Joule heating?
The current density would be inuniform in the copper cylinder. How did you compute the graded current density?
As the current of LTS magnet is cut off within 2 s, the field changes very fast. Do you need to consider the skin effect in the copper cylinder?
The current density of the magnetic dam is shown in Figure 6, but we did not consider Joule heating in the simulation, because the change of resistance and the change of Young's modulus with temperature will make the problem extremely complicated.
As mentioned in the previous answer, the skin effect of the copper cylinder is not considered in the current calculation of the HTS magnet; however, as can be seen in Figure 6, the current distribution of the magnetic dam is not uniform in the finite element simulation. It is related to changes in the magnetic field.