Resistance
3.2 RESISTANCE: CIRCULAR WIRES
For a circular wire, the quantities appearing in Eq. (3.1) are defined by Fig. 3.2. For two wires of the same physical size at the same tempera- ture, as shown in Fig. 3.3(a),
the higher the resistivity, the more the resistance.
As indicated in Fig. 3.3(b),
the longer the length of a conductor, the more the resistance.
Figure 3.3(c) reveals for remaining similar determining variables that the smaller the area of a conductor, the more the resistance.
Finally, Figure 3.3(d) states that for metallic wires of identical con- struction and material,
the higher the temperature of a conductor, the more the resistance.
For circular wires, the quantities of Eq. (3.1) have the following units:
r: CM-ohms/ft at T20°C l: feet
A: circular mils (CM) Rr
A l
l
T (°C) A FIG. 3.2
Factors affecting the resistance of a conductor.
R1 Copper
R2 Iron
R1 Copper R2 Copper
R1 Copper
R2 Copper
R1 Copper
R2 Copper
l2 > l1 (b)
A2 < A1 (c)
T2 > T1
(d) 2 > 1
(a)
FIG. 3.3
Cases in which R2>R1. For each case, all remaining parameters that control the resistance level are the same.
G
Note that the area of the conductor is measured in circular mils (CM) and not in square meters, inches, and so on, as determined by the equation
rradius
ddiameter (3.2) The mil is a unit of measurement for length and is related to the inch by
1 mil in.
or 1000 mils 1 in.
By definition,
a wire with a diameter of 1 mil has an area of 1 circular mil (CM), as shown in Fig. 3.4.
One square mil was superimposed on the 1-CM area of Fig. 3.4 to clearly show that the square mil has a larger surface area than the cir- cular mil.
Applying the above definition to a wire having a diameter of 1 mil, and applying Eq. (3.2), we have
Therefore,
1 CM sq mils (3.3a)
or 1 sq mil CM (3.3b)
Dividing Eq. (3.3b) through will result in
1 sq mil CM1.273 CM
which certainly agrees with the pictorial representation of Fig. 3.4. For a wire with a diameter of N mils (where N can be any positive number),
A sq mils
Substituting the fact that 4 /p CM 1 sq mil, we have
A (sq mils) CM
N2CMSince d N, the area in circular mils is simply equal to the diame- ter in mils square; that is,
(3.4) Verification that an area can simply be the diameter squared is provided in part by Fig. 3.5 for diameters of 2 and 3 mils. Although some areas are not circular, they have the same area as 1 circular mil.
ACM(dmils)2 p4 pN2 4 pN2
4
pN2 4 pd2
4 4p
4p p
4 A pd2
4 p
4 p
(1 mil)2 4 sq mils 1 CM
by definition
1 1000 Area (circle) pr2 p
4 d2
RESISTANCE: CIRCULAR WIRES 61
FIG. 3.4
Defining the circular mil (CM).
1 mil
1 circular mil (CM) 1 square mil
A = (2 mils)2 = 4 CM
1 2
3 4
3
A = (3 mils)2 = 9 CM 2 1
4
5
7 8
6 9
d = 2 mils d = 3 mils
FIG. 3.5
Verification of Eq. (3.4): ACM(dmils)2.
In the future, therefore, to find the area in circular mils, the diameter must first be converted to mils. Since 1 mil 0.001 in., if the diameter is given in inches, simply move the decimal point three places to the right. For example,
If the diameter is in fractional form, first convert it to decimal form and then proceed as above. For example,
in. 0.125 in. 125 mils
The constant r(resistivity) is different for every material. Its value is the resistance of a length of wire 1 ft by 1 mil in diameter, measured at 20°C (Fig. 3.6). The unit of measurement for rcan be determined from Eq. (3.1) by first solving for rand then substituting the units of the other quantities. That is,
r
and Units of r CM
ft
•
The resistivity r is also measured in ohms per mil-foot, as deter- mined by Fig. 3.6, or ohm-meters in the SI system of units. Some typi- cal values of rare provided in Table 3.1.
EXAMPLE 3.1 What is the resistance of a 100-ft length of copper wire with a diameter of 0.020 in. at 20°C?
Solution:
r10.37 0.020 in. 20 mils
ACM(dmils)2(20 mils)2400 CM
Rr
R2.59
EXAMPLE 3.2 An undetermined number of feet of wire have been used from the carton of Fig. 3.7. Find the length of the remaining cop- per wire if it has a diameter of 1/16 in. and a resistance of 0.5 . Solution:
r10.37 CM⋅/ft in. 0.0625 in. 62.5 mils ACM(dmils)2(62.5 mils)23906.25 CM
Rr ⇒ l
l188.34 ft
1953.125 10.37 (0.5 )(3906.25 CM)
10.37 CM
ft ⋅ RAr
l A
1 16
(10.37 CM⋅/ft)(100 ft)
400 CM
l A
CM⋅ ft
AR l 1
8
0.02 in. 0.020 mils 20 mils
TABLE 3.1
Resistivity (r) of various materials.
Material r@ 20°C
Silver 9.9
Copper 10.37
Gold 14.7
Aluminum 17.0
Tungsten 33.0
Nickel 47.0
Iron 74.0
Constantan 295.0
Nichrome 600.0
Calorite 720.0
Carbon 21,000.0
1 mil
1 ft
FIG. 3.6
Defining the constant r(resistivity).
Aklae dry;ketlk sga thrjdrhert dftght tew tij mwet trju ryrt wtyuhw rotjuiks reyt jkur weryty sdfgsg wer ijerw ryrt wtyuhw dfjghfgklil reyhery etyikerwyh y dfjghfgjhkil rotjuiks reyt jkur weryty rstulpio wer ijerw ryrt wtyuhw tdhyhgkrdr
Aklae dry;ke tew tij mwet t rotjuiks reyt jk wer ijerw ryrt wty reyhery etyikerw rotjuiks reyt jk wer ijerw ry Aklae dry;k
tew tij m wet tr rotjuiks reyt jkur wer ijerw
ryrt reyhery etyiker rotjuiks re
FIG. 3.7 Example 3.2.
G
EXAMPLE 3.3 What is the resistance of a copper bus-bar, as used in the power distribution panel of a high-rise office building, with the dimensions indicated in Fig. 3.8?
Solution:
5.0 in. 5000 mils in. 500 mils
ACM A (5000 mils)(500 mils) 2.5 106sq mils 2.5 106sq mils
A3.185 106CM
Rr
R9.768 106
(quite small, 0.000009768 )
We will find in the chapters to follow that the less the resistance of a conductor, the lower the losses in conduction from the source to the load. Similarly, since resistivity is a major factor in determining the resistance of a conductor, the lower the resistivity, the lower the resis- tance for the same size conductor. Table 3.1 would suggest therefore that silver, copper, gold, and aluminum would be the best conductors and the most common. In general, there are other factors, however, such as malleability (ability of a material to be shaped), ductility (ability of a material to be drawn into long, thin wires), temperature sensitivity, resistance to abuse, and, of course, cost, that must all be weighed when choosing a conductor for a particular application.
In general, copper is the most widely used material because it is quite malleable, ductile, and available; has good thermal characteristics;
and is less expensive than silver or gold. It is certainly not cheap, how- ever. Wiring is removed quickly from buildings to be torn down, for example, to extract the copper. At one time aluminum was introduced for general wiring because it is cheaper than copper, but its thermal characteristics created some difficulties. It was found that the heating due to current flow and the cooling that occurred when the circuit was turned off resulted in expansion and contraction of the aluminum wire to the point where connections could eventually work themselves loose and dangerous side effects could result. Aluminum is still used today, however, in areas such as integrated circuit manufacturing and in situa- tions where the connections can be made secure. Silver and gold are, of course, much more expensive than copper or aluminum, but there are places where the cost is justified. Silver has excellent plating character- istics for surface preparations, and gold is used quite extensively in integrated circuits. Tungsten has a resistivity three times that of copper, but there are occasions when its physical characteristics (durability, hardness) are the overriding considerations.