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Thermal Runaway in Electrical Discharges (Electrical Engineering)

In high-voltage transmission substation equipment, the leakage current intensity II (in milliamperes) across a deteriorating ceramic insulator gap is modeled as a function of the applied breakdown electrical stress VV (in kilovolts) by the rational expression:

I(V)=−2.5V−12I(V) = \frac{-2.5}{V - 12}

If localized voltage surges approach the critical breakdown dielectric rating of Vc=12 kVV_c = 12\text{ kV} from an overvoltage condition (V→12+V \to 12^+), substation protection engineers need to map the polarity and magnitude trajectory of the energy discharge.

  1. Classify the behavior of I(V)I(V) as V→12+V \to 12^+ using directional sign analysis.
  2. Evaluate the limit analytically:
    lim⁡V→12+−2.5V−12\lim_{V \to 12^{+}} \frac{-2.5}{V - 12}
  3. Explain how the resulting vertical asymptote informs arc-flash circuit breaker trigger ratings.