How to calculate the current carrying capacity of copper and aluminium cable?

Officially, you have to look it up, in the National Electric Code, if you are going to build anything permanent out of it. You will find plenty of online wire sizing calculators that will do it for you, given the maximum continuous current, the operating current, and the operating voltage. http://phd.boschsecurity.com/wiregage_cal.cfm --------------------------------------... But the way that it is calculated there, is that it is a mixed thermal and electric problem. There is other experience that is incorporated in to the NEC tables, that can overlook the following, but the following does serve as a baseline for how it is done. The condition that must be met, is that the temperature of the insulation MUST NOT exceed a certain limit, based upon how much temperature the insulation is rated to withstand. Most insulation is rated at about 75 Celsius. Some insulation is rated at 90 Celsius, for when you are within the bundle in the long run of conduit, but locally at the terminals, you still need to make sure it doesn't exceed 75 C. Within the wire, there is heat generated due to the current flowing with resistance per unit length. As a result, it generates heat per unit length, which must be dissipated out through the electrical insulation. The insulation is only intended to be electrically insulating, but inevitably, it is also thermally insulating. Therefore, it has a temperature differential from inside to outside, which must not exceed the insulation temperature rating minus the background temperature. The wire itself must not exceed certain limitations for the copper or aluminum. I.e. it must not get hot enough to melt, and you must stay away from this within a certain safety margin. There is a thermal problem, of how to calculate temperature rise from surface to core, within a significantly long cylinder that is actively generating heat within. Besides just the LOCAL ampacity problem of making sure that insulation and conductor do not overheat from the maximum expected current, there is also the cumulative problem, of making sure that the voltage drop due to the resistive heating at the operating current, doesn't exceed a certain percentage of the operating voltage. Generally, within an entire site of one given utility service, voltage drop of any given entire system should be kept to not exceed 3% of the operating voltage. It isn't a big deal if it gets to be 3.1%, but it should be planned not to exceed 3%, to avoid inconsistent operation within any of the components. Simply enough, both the ampacity problem, and the voltage drop problem, are to be solved "in parallel". Which ever yields the larger size needed, is the size of wire you select.