Encyclopedia of Electrical Engineering

Since temperature can have such a pronounced effect on the resistance of a conductor, it is important
that we have some method of determining the resistance at any temperature
within operating limits.
An equation for this purpose can be obtained by approximating the curve (in Fig.no.1 ) by the straight
dashed line that intersects the temperature scale at -234.5℃.
Although the actual curve extends to absolute zero (-273.15℃, or 0 K),
the straight-line approximation is quite accurate for the normal operating
temperature range. At two temperatures T1 and T2, the resistance
of copper is R1 and R2, respectively, as indicated on the curve.
Using a property of similar triangles, we may develop a mathematical
relationship between these values of resistance at different temperatures.
Let x equal the distance from -234.5℃ to T1 and y the
distance from -234.5℃ to T2, as shown in Fig.No.1 From similar
triangles,
$$ {x \over R1} = {y \over R2}$$
$$\bbox[5px,border:1px solid blue] {\color{blue}{{234.5 + T1 \over R1} = {234.5 + T2 \over R2}}} \tag{1}$$
**Fig.no.2: **Similar Triangles have equal ratio of sides.
The temperature of -234.5℃ is called the inferred absolute temperature
(Ti) of copper. For different conducting materials, the intersection
of the straight-line approximation occurs at different temperatures. A
few typical values are listed in Table 1.
The minus sign does not appear with the inferred absolute temperature
on either side of Eq. (1) because x and y are the distances from
-234.5℃ to T1 and T2, respectively, and therefore are simply magnitudes.
Eq. (1) can easily be adapted to any material by inserting the
proper inferred absolute temperature. It may therefore be written as
follows:
$$\bbox[5px,border:1px solid blue] {\color{blue}{{|T_i| + T1 \over R1} = {|T_i| + T2 \over R2}}} \tag{2}$$
where $|T_i|$ indicates that the inferred absolute temperature of the material
involved is inserted as a positive value in the equation. In general,
therefore, associate the sign only with T1 and T2. The temperature of -234.5℃ is called the inferred absolute temperature (Ti) of copper. For different conducting materials, the intersection
of the straight-line approximation occurs at different temperatures. A
few typical values are listed in Table 1.
**Table.no.1: **Inferred absolute temperatures (Ti).
**Example 1:** If the resistance of a copper wire is 50 Ω at 20℃, what
is its resistance at 100℃ (boiling point of water)?

**Solution:** According to Eq. (1):
$$ {234.5℃ + 20℃ \over 50 Ω} = {234.5℃ + 100℃ \over R2}$$
$$R2 = {(50 Ω)(334.5℃) \over 254.5℃} = 65.72 Ω$$

Resistance: Circular Wires
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Temperature Coefficient of Resistance