Power in Electric Circuits — Question 8

Question 8

An adiabatic fuse model converts a current pulse into a melting criterion

A short wire has mass m, specific heat c, melting temperature $T_m$, initial temperature $T_0<T_m$, and latent heat of fusion $L_f$. Its resistance R is treated constant; heat loss is neglected.

1. Find the time to melt completely under constant current I.

2. For $I(t)=I_pe^{-t/\tau}$, find the minimum peak current that supplies enough energy over all time.

Question 8 - Solution

An adiabatic fuse model converts a current pulse into a melting criterion

(a) Method and calculation

$$W_m=mc(T_m-T_0)+mL_f.$$$$I^2Rt_m=W_m\Rightarrow t_m=\frac{m[c(T_m-T_0)+L_f]}{I^2R}.$$

(b) Interpretation and further calculation

$$W_\infty=R\int_0^\infty I_p^2e^{-2t/\tau}dt=RI_p^2\tau/2.$$$$I_p\ge\sqrt{2W_m/(R\tau)}.$$Equality approaches complete melting only as time tends to infinity.

Physical check

The threshold is an ideal energy criterion; real resistance and heat losses vary during heating.