The resistance to the flow of direct current (DC) of a circular solid conductor can be calculated with precision if three values are known:
The resistance R can be calculated from:
R = ρ (l/A) where: ρ is the electrical resistivity in Ohm-meters l is the length in meters, and A is the cross section area in meters2
The formula (above) describes an easily understood relationship between the properties of the conductor and its electrical resistance. The resistance increases with increasing length of the conductor and decreases with increasing cross-section area. And the resistance is directly proportional to the properties of the material of the conductor.
Increased resistance with increased length is easily understood, and perhaps needs no further elaboration. As is often the case in scientic definitions, the dimensions for length are in meters
Decreased resistance with increased cross-section can be understood by using an analogy of water flowing through a pipe. If the flow rate of the water is analogous to electrical current flow, a pipe of larger diameter can be understood to present less obstruction to the flow of water at a certain rate than would occur in a much smaller diameter pipe. The dimensions of the cross-section area are in meters2.
The pipe and water analogy can also be useful in understanding how the conductor's material affects resistance to current flow by imagining a pipe that was completely clear and empty compared to one with residue or clogs inside. If you have fought with a slow shower drain, you understand this analogy. The dimension of resistivity are in units of Ohm-meters; this is by design so that the dimensional units in the formula cancel and leave only Ohms as the unit.
The degree to which a material resists the flow of electrical current is called its resistivity. For common materials used as electrical conductors, the value of their electrical resistivity is well known and defined.
The most common electrical conductor is the element copper. There is an interesting history to the establishment of a standard value for the resistivity of copper. A value was set in 1913, called the International Annealed Copper Standard (often written just as IACS), based on measurements of the annealed copper that was produced by 14 different refiners of that era. The term annealed refers to the treatment or working of the copper to prepare it for being drawn through dies to make circular wire conductors. (Annealing copper affects its resistivity, increasing it very slightly from pure copper that has not been annealed.) The accepted value for IACS resistivity is now ρ = 1.7241x10-8 Ohm-meter. Because resistivity varies with temperature, this value is for copper at 20°C.
To demonstrate how the formula works, the resistance of a 10-AWG annealed copper conductor will be calculated and compared to a standard value listed in wire tables. The sizing of electrical conductors by a wire gauge system is an artifact of how wire was (and still is) produced: copper was drawn through various dies to extrude it into a circular conductors. Each subsequent pass through a die reduces the wire diameter created. The gauge of a wire is related to the number of dies it has been drawn through: the more dies, the smaller the wire and the higher number for its gauge. Thus a wire of 10 American Wire Gauge or 10 AWG would have been pulled through two more dies than a wire of 8 AWG. (A Wikipedia article explains how AWG wire sizes came into being in detail).
For calculation the length of the conductor will be 1,000 feet: wire resistance per 1,000 feet is a common value provided in tables, so we can have a value to compare to our calculated value. We need to convert to meters for calculation: l = 3.048 × 102 meters.
Rather than attempt to derive the diameter of a 10-AWG wire by calculation, a value will be taken from a wire size table: 0.1019 inch. (See below for a method to calculated diameter from wire gauge.) Conversion to meters results in d = 2.58826 × 10−3 meters. To get the radius r we divide d by two, then compute the cross-section area from:
A = πr2 A = π × 1.6747 × 10−6 meters A = 5.2609 × 10−6 meters
Finally, we enter the values for resistivity, length, and cross-section into the formula for Resistance:
R = ρ(l/A) R = (1.7241 x 10-8 × 3.048 × 102) / 5.2609 × 10−6) R = 5.2550568 × 10−6 / 5.2609 × 10−6 R = 0.9989 Ohms
The value for 10-AWG wire resistance in tables is usually given as 0.9989 Ohms; the value calculated here is exactly the same. (The table value appears to have been calculated by using no more than four decimal places in any of the values.) Because copper's resistivity changes with temperature, the resistance of the conductor will change: resistance increases with temperature.
The flow of an alternating current in a conductor will be affected by a property called skin effect which results in an uneven distribution of the current flow, concentrating it at the outer surface of the conductor. For a thorough treatment see the Wikipedia article on Skin Effict.
An expression for finding the diameter d in inches of a conductor of size AWG n is:
d = 0.005 inch × 92[(36−n)/39]
Evaluating at n =10, we find:
d = 0.005 inch × 92[(36−10)/39] d = 0.005 inch × 92[26/39] d = 0.005 inch × 920.666 d = 0.005 inch × 20.3794 d = 0.1019 inch
The value from tables is 0.1019, and the calculated value provides good agreement.
An expression for finding the diameter d in mm of a conductor of size AWG n is:
d = 0.127 mm × 92[(36−n)/39]
Evaluating at n =10, we find:
d = 0.127 mm × 92[(36−10)/39] d = 0.127 mmh × 92[26/39] d = 0.127 mm × 920.666 d = 0.127 mm × 20.3794 d = 2.588 mm
The value from tables is 2.588, and the calculated value provides good agreement.
Copyright © 2018 by James W. Hebert. Unauthorized reproduction prohibited!
Author: James W. Hebert
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This article first appeared January 17, 2018.