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Time-dependent cathode potential drop (CPD) is studied for a transient cathode spot on a Cu cathode protrusion when the spot operates continuously. A transient heat model was self-consistently developed, taking into account the kinetics of cathode vapor flow in a Knudsen layer, considering the momentum and energy conservation in the sheath, the electron relaxation zone, and the hydrodynamic region of the expanding plasma. The difference between evaporated and return fluxes of heavy particles determines the protrusion erosion rate and, hence, the decrease of its size during transient spot operation. It is shown that the spot parameters (e.g., CPD, temperature, and density) change from an initial to a steady-state value with time. Thus, for continuous spot operation on a Cu protrusion with a 5-μm radius, the CPD is about 90 V when the spot originates from initial plasma with a time duration of less than 50 ns and decreases to 13 V when spot operation continues up to 1 μs. This result explains the relatively large power supply voltage (~100 V) needed for arc ignition (after triggering the initial plasma at the cathode in a low-voltage electrode gap) compared to the low CPD appearing during steady-state arc operation. The calculated potential drop in the plasma expansion region at steady state is about 5 V, and the sum of this value with the CPD agrees well with the measured Cu arc voltage. The spot initiation and development can be consistently described as a vaporization mechanism by the gasdynamic model without involving any phenomena of explosion of irregularities at the cathode surface.