Superconductors — Question 4

Question 4

Temperature margin limits a superconducting tape current

A tape has a supplied engineering critical-current-density law $J_c(T)=J_{c0}[1-(T/T_c)^2]$ for $0\le T<T_c$. The engineering area A is fixed. It operates at current $I=.60AJ_{c0}$, and the admissible condition is $I<AJ_c(T)$.

1. Find the largest allowed temperature as a fraction of Tc.

2. At $T=.50T_c$, find the fractional current margin $(I_c-I)/I_c$ and the increase in critical current if temperature is lowered to $.30T_c$.

Question 4 - Solution

Temperature margin limits a superconducting tape current

(a) Method and calculation

$$.60<1-(T/T_c)^2\Rightarrow T/T_c<\sqrt{.40}.$$$$T_{\mathrm{boundary}}=.632456T_c\quad\text{(operate below this boundary)}.$$

(b) Interpretation and further calculation

$$I_c(.50T_c)=.75AJ_{c0},\quad\frac{I_c-I}{I_c}=\frac{.75-.60}{.75}=.200.$$$$I_c(.30T_c)/I_c(.50T_c)=.91/.75=1.21333.$$

Physical check

This is an area-averaged engineering limit; it does not assume microscopic current density is uniform through the tape.