Difference between short circuit and earth fault current

Both are short circuits. A short circuit is an unintended connection between phases or between on or more phases and earth. The scenario with a connection from one phase to earth is typical what we name an earth fault. An earth fault can occur if you have a 3 phase system in underground cables with 1 phase per cable. If one cable is digged into accidentally you would get an earth fault. Could also be a 3ph system in overhead lines with a tree getting too close to one phase. This would be an earth fault too, perhaps just a temp earth fault. A short circuit with two or more phases and perhaps earth as well could occur in a switchgear with a bus bar fault, a three phase cable fault etc.

When current carrying conductors conduct to non-current carrying conductors such as an enclosure, (no load) the bond wire safely conducts the current to earth ground.
When current carrying conductor contacts another current carrying conductor with no load to restrict current flow this is also a short, because there is no load.
The net result is generally the same during an event; short circuit current flows (low resistance) current spikes and blows fuse or breaker.

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