In electrical engineering, "ampacity" refers to the maximum level of current that a conductor in a particular environment can carry before either the conductor or its insulation melts. To calculate the ampacity of a copper wire, you must use the Neher-McGrath equation, a form of the Fourier heat transfer equation adapted specifically for electrical applications.

- Skill level:
- Moderate

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## Instructions

- 1
Determine the temperature, in degrees Celsius, of the environment where the wire will be located. If the wire is going to be running through the ceiling of a basement, use the air temperature of the basement. If the wire is going to be buried underground, use the highest average ground temperature recorded for the summer.

- 2
Determine the melting point, in degrees Celsius, of the wire's jacket insulation. Consult a list of common insulation materials and their melting points (see Resources).

- 3
Determine the angular frequency of the current that will be running through the wire. In North American, the standard angular frequency for electrical power is 60 hertz.

- 4
Determine the thermal conductivity for copper, the wire insulation and the surrounding environment (e.g. air, dirt, rock, wood or water). Consult a table of thermal conductivity values of common materials (see Resources).

- 5
Convert each of the thermal conductivity value you determined in Step 4 into thermal resistivity values by dividing one by the thermal conductivity value.

- 6
Add together all of the thermal resistivity values you calculated in Step 5. This new value represents the combined thermal resistance of the wire in its environment.

- 7
Determine the length of the wire in kilometres. Note: 1 meter = 0.001 kilometres.

- 8
Determine the resistance rate (in ohms per kilometre) for that wire's particular gauge number. Consult a table of copper wire gauge resistance rates (see Resources).

- 9
Multiply the wire's resistance rate (from Step 8) by its length (from Step 7) to calculate the baseline resistance for the wire.

- 10
Subtract 25 from the melting point of the insulation (from Step 2).

- 11
Multiply the result from Step 10 by 3.9 x 10^-3 /ºC.

- 12
Add one to the result from Step 11, and multiply that sum by the wire's baseline resistance (from Step 9). This new value represents the wire's baseline resistance plus temperature-dependent resistance.

- 13
Multiply the frequency (in hertz) of the electrical current (from Step 3) by 3.94 x 10 ^-6.

- 14
Divide the result from Step 12 by the result from Step 13.

- 15
Take the square root of the result from Step 14. This value is known as the "skin depth" of the wire.

- 16
Multiply the skin depth (from Step 15) by the cross-sectional diameter of the wire, in inches. Once you've completed this calculation, multiply the result by 3.14 (an approximation of pi).

- 17
Multiply the length of the wire (from Step 7) by 39370.079 inches/kilometre to convert its units from kilometres to inches.

- 18
Divide the inch length of the wire (from Step 17) by the result from Step 16.

- 19
Multiply the result from Step 18 by the result from Step 11 (i.e. the wire's baseline resistance plus temperature-dependent resistance). The value you calculate will represent the resistance of the entire wire when its temperature is just below the insulation's melting point and its load has the angular frequency you specified in Step 3.

- 20
Subtract the environment temperature (from Step 1) from the insulation's melting point (from Step 2). Call this the "temperature gradient."

- 21
Multiply the total wire resistance (from Step 19) by the combined thermal resistance (from Step 6).

- 22
Divide the temperature gradient (from Step 20) by the result from Step 21.

- 23
Take the square root of the result from Step 22. The positive root represents the ampacity of your copper wire, in kiloamperes (kA).