step0: fully shielded
by the barrier
step1: partial penetration
in the barrier
step2: full penetration
through the barrier
step3: start penetration
into the filament
step4: most of the
step5: full penetration
in to filaments
||The above simulation is still imperfect. For example, the penetration of magnetic flux has to be zero in total; persistent current should be internal. Improvement in this simulation is the forcing of symmetric penetration. It looks like almost prefect in the flux distribution. Now the calculation of the magnetization should be possible.|
|Let me start with a slab model. It is easier to compare with the analytical model. The magnetization behavior of the model turned out to be close to the theoretical model. Still there are some glitches at the turning points of the field. This is due to the blanked field sensitivity introduced to model the system in the finite element method. There could be some more improvement.|
|These two curves are the magnetization of DT conductor with and without barrier magnetization based on the test geometry of 7-strand. The field dependence of Jc is not included yet. The two-step magnetization coming from the barrier effect is clearly seen in the curve and the increase of loss due to the barrier is evident.|
|A simple field dependence of jc=jc0*4/(2+B) was introduced in this simulation for a 7-filament conductor with and without barrier. The model is getting close to the reality. The magnetization shape and the difference between Ta barrier and Nb barrier are evident. By the use of real geometry, it should be possible to evaluate the loss of the real DT conductor. It will just take more CPU time.|
|The flux behavior is best visualized by a motion picture. Click the left icon to see the lux movie. The penetration of the flux into the virgin state and the re-distribution of the shielding current for the decrease of the field is in the show.|
|Although magnetization curve would not be much affected, the symmmetric penetration is a little too much restriction for the flux. the symmmety axis has to be at least tilted at near the barrir. Improvement in this picture is the introduction of asymmmetric penetration. Now the persistent current is zero in total in each filament and yet the penetration into the filament is not forced to be symmmetric. It looks like more natural.|
The magnetization of N filament with radius
a can be calculated as the sum of current moment:
, where jc is the current density in the filament. The magnetization is measured par volume base. The volume of the conductor is:
, where eta is the non-superconductor to superconductor ratio. Then the measured magnetization, using Jc(over all current density including copper stabilizer) is:
Effective diameter of the conductor is often evaluated using this equation.
We have used this equation in our publication. However, if the measurement system is calibrated against a niobium cylinder, we should have used
If the calibration was made using Nb wire of the same size, eq.(4) is correct. The magnetization of the barrier with outer radius R and inner radius r is:
If we measure the conductor with no filament,
is the volume and
would give the measured magnetization par volume. This expression should be effective for tube method conductor or PIT conductor.
R=350u a=25u r=354(effective with reduced jc) in our case,
Mb and M becomes about the same. Since the filament magnetization is reduced by the shielding effect of the barrier, total magnetization would be a little less than the sum of the filament and the barrier magnetization. This agrees with our measurement assuming that most of the discrepancy between geometrical size of the filament bundle and the effective diameter comes from the barrier magnetization.
If the diameter of the strand is reduced to 0.7mm the barrier magnetization will be greatly reduced. At diameter 1mm, the use of tantalum barrier is a good way of reducing magnetization.