Mathematical Approach to Current Sharing Problem of Superconducting Triple Strands

The current sharing between insulated strands in a superconducting cable is one of the important problems for its utilization. From the view points of the inverse problem, the sensitivity of current sharing between the insulated strands is determined by the condition number of the inductance matrix. For the triple strands with the self similar structure, we derive the analytic form of the inductance matrix which only includes two parameters; the self inductance of a unit wire, the ratio of mutual to self inductance for unit wires. Since the matrix elements also have the self similar structure, we can analytically obtain the eigenvalues, eigenvectors and condition number, which is the ratio of maximum and minimum eigenvalues. Next, we derive the formula to estimate the sensitivity of the current distribution against the displacement of inductance from the ideal case by use of the condition number. This formula shows that the sensitivity is inversely proportional to the difference of self and mutual inductances of unit wires. Moreover, we estimate the condition number of the very thin wire to check our formula. Finally, we verify our analytic form by numerical calculations.