Many conductors are much less than an inch in diameter. To simplify area calculations for such conductors, the foot-pound system uses an area unit, called a circular mil, based on a mil, or one-thousandth of an inch (see** Figure 1**).

Definition:One circular mil (CM) is the area of a circle 1 mil in diameter.

**Figure 1:** A circle with the diameter of one mil

When using circular mils, the formula for the area of a circle becomes

$\begin{matrix} A={{d}^{2}} & {} & \left( 1 \right) \\\end{matrix}$

Where A is the area of the circle in circular mils and d is the diameter in mils.

**Wire Size Example**

Find the cross-sectional area of a wire 0.0280″ in diameter.

**Solution**

$\begin{align} & d=0.02800”=28\text{ }mills \\ & A={{d}^{2}}={{28}^{2}}=784\text{ }CM \\\end{align}$

**Resistivity Definition**

**Resistivity** can be defined in terms of a piece of wire 1 ft long with a cross- sectional area of 1 CM, as shown in **Figure 2**. The volume of this wire is 1 circular-mil foot, and the units of ρ become CM-Ω/ft. **Table 1** lists the resistivities for some metals.

**Figure 2** One circular-mil foot

Material | Resistivity |

Silver | 9.56 |

Copper | 10.4 |

Gold | 14.0 |

Aluminum | 17.0 |

Tungsten | 34.0 |

Brass | 42.0 |

Iron | 61.0 |

Nichrome | 675.0 |

**Table 1** Specific Resistivities (${}^{\Omega }/{}_{c.mil-ft}$ at 20 ^{o}C)

**Resistance Calculation Example**

Calculate the resistance of 50.0 miles of copper wire 0.3500 in diameter.

** Solution**

$\begin{align} & d=0.350”=35mils \\ & A={{d}^{2}}=122.5CM \\ & l=50miles\times \frac{5280ft}{1mile}=2.54\times {{10}^{5}}ft \\ & R=\rho \frac{l}{A}=\frac{10.36\times \left( 2.54\times {{10}^{5}} \right)}{122.5\times {{10}^{3}}}=22.4\Omega \\\end{align}$

**American Wire Gauge System**

Definition:The system of notation for measuring the size of conductors or wires.

In North America, the most wire is produced in the standard American Wire Gauge (AWG) sizes listed in Table 2. The larger the gauge number, the smaller the wire size.

Wires larger than 4/0 (sometimes called 0000) are usually specified in thousands of circular mils (MCM) instead of having a gauge number.

The ratio of the diameter of adjacent American Wire Gauge sizes is about 1.1229, and the ratio of their areas is about 1.26.

The area ratio between wire sizes three AWG numbers apart is very close to 2. **For example**, #10 wire (10 380 CM) has about twice the area of #13 wire (5178 CM) and half the area of #7 wire (20 820 CM).

Since resistance is inversely proportional to area, the resistance per unit length also differs by a factor of about two for wires three AWG numbers apart. **For example**, #1 wire (0.1239 Ω/1000’) has half the resistance of #4 wire (0.2485 Ω/1000’) and twice the resistance of #3/0 wire (0.0618 Ω/1000’).

**Wire Size Calculation Example**

What size of aluminum wire has a resistance of 1.62 Ω for a length of 2500’?

**Solution**

Since,

\[R=\rho \frac{l}{A}\]

So,

\[A=\rho \frac{l}{R}=17\times \frac{2500}{1.62}=26.2MCM\]

From the **Table 2**, we see that this area matches **AWG #6.**

Gauge Number | Diameter (Mils) | Circular Mil Area | Ohms Per 1000 Ft |

00 | 365 | 133,000 | 0.0795 |

0 | 325 | 106,000 | 0.100 |

1 | 289 | 83,700 | 0.126 |

2 | 258 | 66,400 | 0.159 |

3 | 229 | 52,600 | 0.201 |

4 | 204 | 41,700 | 0.253 |

5 | 182 | 33,100 | 0.319 |

6 | 162 | 26,300 | 0.403 |

7 | 144 | 20,800 | 0.508 |

8 | 128 | 16,500 | 0.641 |

9 | 114 | 13,100 | 0.808 |

10 | 102 | 10,400 | 1.02 |

11 | 91 | 8,230 | 1.28 |

12 | 81 | 6,530 | 1.62 |

13 | 72 | 5,180 | 2.04 |

14 | 64 | 4,110 | 2.58 |

15 | 57 | 3,260 | 3.25 |

16 | 51 | 2,580 | 4.09 |

17 | 45 | 2,050 | 5.16 |

18 | 40 | 1,620 | 6.51 |

19 | 36 | 1,290 | 8.21 |

20 | 32 | 1,020 | 10.4 |

21 | 28.5 | 810 | 13.1 |

22 | 25.3 | 642 | 16.5 |

23 | 22.6 | 509 | 20.8 |

24 | 20.1 | 404 | 26.2 |

25 | 17.9 | 320 | 33 |

26 | 15.9 | 254 | 41.6 |

27 | 14.2 | 202 | 52.5 |

28 | 12.6 | 160 | 66.2 |

29 | 11.3 | 127 | 83.4 |

30 | 10.0 | 101 | 105 |

31 | 8.9 | 79.7 | 133 |

32 | 8.0 | 63.2 | 167 |

33 | 7.1 | 50.1 | 211 |

34 | 6.3 | 39.8 | 266 |

35 | 5.6 | 31.5 | 335 |

36 | 5.0 | 25.0 | 423 |

**Table 2** American Wire Gauge (AWG) Wire Sizes Chart