If a ground rod is the only grounding electrode at a service and the GEC becomes loose, the branch-circuit breakers will take what amount of time to open during a ground fault?

The correct reasoning for why the branch-circuit breakers will take the same amount of time to open during a ground fault even if the grounding electrode conductor (GEC) becomes loose lies in the function and operation of the breakers and the nature of the fault current.

When a ground fault occurs, the branch-circuit breakers are designed to sense an overcurrent condition and operate accordingly. Their response time is determined by their trip curve and the magnitude of the fault current. If the GEC is loose, this may affect the grounding system's overall effectiveness, but it does not directly change the way the circuit breakers respond to the overcurrent created by the ground fault. The breakers are calibrated to trip within specific parameters, and if those thresholds are reached regardless of the grounding system's status, they will operate as intended.

Thus, the timing for the breakers to trip remains unaffected by the condition of the GEC. This is why the correct answer is that they will take the same amount of time to open during a ground fault, emphasizing the resilience of the breakers' design independent of grounding electrode conditions.