A conductor presents a resistance to any current it carries. The "better" the conductor (silver is the best, copper and aluminium are also good), the less this resistance, and the lower the "voltage drop" - the resistance multiplied by the current.
This calculator takes only resistive losses into account, and ignores reactive impedances.
A cable's construction sets a limit on the maximum temperature that the conductors may reach. When operating a cable at an ambient temperature different from that in the conditions used for specifying its current-carrying capacity, you need to apply a derating factor to reduce (or increase!) the current that the cable can safely carry.
This calculator takes only the resistive heating in a cable into account, and makes no attempt to deal with hotspots.
On short time scales, approximately all the power dissipated in a current-carrying conductor remains in the system as heat. This calculator gives the time required to reach a given temperature rise, assuming adiabatic conditions.
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