We investigated the voltage applicable to $Au/YBa_2Cu3O_7$ (YBCO) meander lines. The meander line was fabricated by patterning Au/YBCO thin films grown on sapphire substrates by photolithography. It was subjected to simulated AC fault currents, and the resistance was measured and analyzed. The samples were immersed in liquid nitrogen during the experiment for effective cooling. The voltage applicable to the meander lines depended on the fault duration. Dependence was strong at short fault durations, and weak at long durations. When the voltage was plotted as a function of the fault duration on a log-log scale, data fell more or less on straight lines for all meander lines. In other words, the voltage applicable to Au/YBCO meander lines on sapphire substrates was inversely proportional to $t^b$, where t is the fault duration and b ranges from 0.4 to 0.5. The results were analyzed quantitatively with the concept of heat balance. Under adiabatic condition, the voltage is to be inversely proportional to $t^{0.5}$ for all samples. Less value of b for some samples is thought to be due to cooling of the samples by liquid nitrogen.