A Study on the Temperature Co-efficient of Resistance

Temperature plays a crucial role in electrical conducting materials. The conductivity of conductors is affected when the temperature changes, though this is not always the case. In this article, a detailed explanation is given about how temperature affects conductivity in electrical engineering materials. The temperature co-efficient of resistance plays a vital role in the behavior of these materials

When temperature changes, the resistance of most electrical engineering materials also changes. In some conductors, resistance increases, while in others, it decreases. However, there are a few conductors whose resistance remains constant, irrespective of temperature changes. Based on these changes, materials are classified as follows

In electrical engineering, copper, aluminum, iron, and silver are commonly used as conducting materials because they have a very high conductance range. Conductors with a small cross-sectional area can carry more current than others. As temperature increases, their resistance also increases in a linear manner up to a certain temperature range. Beyond this point, these conductors no longer follow linear behavior

Alloys are generally known as compounds made from two or more metals. Manganin and Eureka are commonly used to fabricate heating elements. In alloys, resistance does not increase as much as it does in pure metals—the resistance-temperature curve is almost flat. This is why alloys are preferred for making heating elements

The resistance of electrolytes decreases as temperature increases. This can be observed when the temperature of a chemical cell rises—it discharges more rapidly. Insulators can become conductors when the temperature increases. Carbon lies somewhere between conductors and insulators; we refer to it as a semiconductor. With a considerable increase in temperature, carbon behaves like a conductor

To study the temperature coefficient, we consider a pure metal such as copper, which is popularly used as a conducting material in electrical engineering applications. As we have seen earlier, when temperature increases, the resistance of pure metals also increases. Let us consider the temperature changing between 0°C and 100°C. The changes in resistance then depend on

1. Proportional to the initial temperature of the material
2. Proportional to the change in temperature.
3. Depends on the nature of the material.

Let us consider that,

Initial Resistance at 0°C is R₀ Ohm ((Ω)

When it heated to t°C , Resistance increases to R_t Ohm ((Ω)

Changes in Resistance = Increased Resistance – Initial Resistance

Then, changes in resistance ∆R = R_t – R₀

This changes in resistance proportional to initial resistance and temperature

R_t – R₀ α R₀ x t

R_t – R₀ = α₀ R₀ x t

By applying the constant α_0​, which is known as the temperature coefficient of resistance at 0°C, we can analyze resistance changes. This value depends on the nature of the conductor material. To derive the temperature coefficient of resistance from the above equation

\[ \alpha_0 = \frac{R_t – R_0}{R_0.t}\]

Temperature co-efficient of resistance defined at 0°C defined as the change in resistance per ohm original resistance per degree centigrade change in temperature from 0°C

(or)

It is the ratio of the change in resistance per degree centigrade change in temperature to the resistance at 0°C

For example , let us consider as

Initial resistance R₀ = 1 Ohm

Changes in Temperature t = 1°C

Then temperature co-efficient of resistance

\[\alpha_0 = \Delta R = R_r – R_0\]

In addition to , temperature co-efficient of resistance can be defined as at 0°C can also be defined as the change in resistance of one ohm at 0°C for 1°C change in temperature.

from equation of temperature co-efficient of resistance, by substituting units for respective parameters

\[ \alpha_0 = \frac{\Delta R}{R_0 . t}\]

\[ \alpha_0 = \frac{ohm}{ohm x ^\circ C}\]

\[ \frac{1}{^\circ C}\]

Hence, the unit of temperature co-efficient of resistance is per degree centigrade.

From equation R_t – R₀ = α₀ R₀ x t

By simplifying above equation

R_t = R₀ + α₀ R₀ x t

R_t = R₀ (1+ α₀ x t)

This expression is more helpful in finding the resistance of the conductor at any temperature provided its initial resistance R0 at 0°C and the constant α₀ are known

According to the nature of temperature coefficient properties, electrical engineering materials can be classified as follows

1. Positive Temperature co-effiecient Materials
2. Negative Temperature co-efficient of materials

Materials whose resistance increases from the initial resistance when the temperature increases, and decreases when the temperature decreases, are called materials with a positive temperature coefficient.

Example : All conducting materials such Copper, Aluminum, Iron, Silver

Materials whose resistance decreases from the initial resistance when the temperature increases, and whose resistance increases when the temperature decreases, are called materials with a negative temperature coefficient

Example: Rubber, PVC, Silicon, Genramium, Carbon

Let us review the formula again, which helps find the resistance of a material at any temperature

R_t = R₀ (1+ α₀ x t)

In this formula

R_t = Resistance in Ohm at temperature t°C

R₀ = Initial resistance in ohm at 0°C

α₀ = Temperature co-efficient of resistance at initial temperature 0°C

t = Increased temperature in °C

It is most important to consider that the temperature coefficient of resistance (α) is not the same for all temperatures. It also differs from one material to another. Each material has a different temperature coefficient, and it changes with temperature. This value depends on the initial temperature from which the temperature starts to increase. In practical cases, normal ambient temperatures accommodate most materials, although some industries use them in both high-temperature and low-temperature applications

The temperature co-efficient of resistance at any given reference temperature can be defined as change in resistance per ohm original resistance per degree centigrade change in temperature from given reference temperature.

It is experimentally observed that, in the higher temperature range, the temperature coefficient of resistance decreases. It is maximum at 0°C in all materials

For instance, to calculate changes in resistance when the temperature increases from a value other than 0°C, let us consider an initial temperature ttt (say 25°C). The temperature then increases to T°C. The change in resistance can be obtained by modifying the above equation.

R_T = R_t (1+ α_t x (T-t))

To study the relationship between the temperature coefficient at different temperatures, we assume that heating a conducting material raises its temperature from an initial T₁​ °C to a new T₂​ °C.

. Consequently, the resistance changes from R_1​ ohms to R_2​ ohms as the temperature changes. Corresponding to these temperature changes, the temperature coefficient of resistance also changes from α₁​ to α_2​

By applying these parameters in the above equation

R₂ = R₁ (1+ α₁ x (T₂-T₁))

On the other hand, a conducting material is cooled from an initial temperature T_2​ °C to a final temperature T_1​ °C. The above equation can be rewritten as

R₁ = R₁ (1+ α₂ x (T₁-T₂))

By equating both of these equation

\[ \frac{R_1}{R_2} = \frac{[1+ \alpha_2 (T_1 – T_2]}{[1+ \alpha_1 (T_2 – T_1]}\]

\[ \alpha_2 (T_2 – T_1 ) = \frac{1}{[1+ \alpha_1 (T_2 – T_1]} – 1\]

\[ \alpha_2 (T_2 – T_1 ) = \frac{\alpha_1 (T_1 – T_2)}{[1+ \alpha_1 (T_2 – T_1]} \]

\[ \alpha_2 = \frac{1}{[1+ \alpha_1 (T_2 – T_1]} \]

(or)

\[ \alpha_2 = \frac{1}{[\frac{1}{\alpha_1} + (T_2 – T_1]}\]

This equation is most helpful for finding changes in the temperature coefficient of resistance from an initial temperature T₁​ °C to a new temperature T₂​ °C.

Sl.No	Name of Material	Temperaure co-efficient of Resistance per °C	Nature of Material
1	Aluminum	0.004041	Positive temperature co-efficient
2	Copper	0.004041	Positive temperature co-efficient
3	Constantan	0.000074	Positive temperature co-efficient
4	Gold	0.003847	Positive temperature co-efficient
5	Iron	0.005671	Positive temperature co-efficient
6	Manganin	0.000015	Almost zero temperature co-efficient
7	Molybdynum	0.004579	Positive temperature co-efficient
8	Nichrome	0.00013	almost zero temperature co-efficient
9	Nickel	0.005866	Positive temperature co-efficient
10	Silver	0.003819	Positive temperature co-efficient
11	Tungsten	0.004403	Positive temperature co-efficient
12	Carbon	-0.0005	Negative temperature co-efficient
13	Germanium	-0.00048	Negative temperature co-efficient
14	Silicon	-0.00075	Negative temperature co-efficient

The DC motor field winding resistance increases from 100 ohms at 25 °C to 120 ohms at 55 °C. Find the temperature coefficient of the winding material at 0 °C.

GIven:

Initial Temperature T1 = 25 °C

Final temperature T2 = 55 °C

Initial resistance R₂₅ = 100 Ohm at 25 °C

Final resistance R₅₅ = 120 Ohm at 55 °C

To find

Temperature co-efficient of material at 0°C

Solution

Formula to find temperature co efficent of resistance at any temperature R_t = R₀ (1+ α₀ x t)

Temperature co-efficient of resistance at 25 °C

R₂₅ =100 = R₀ (1+ α₀ x 25 )

Similarly ,

Temperature co-efficient of resistance at 25 °C

R₅₅ =120 = R₀ (1+ α₀ x 55 )

Dividing R₂₅ by R₅₅

\[\frac{100}{120} = \frac{1+ \alpha_0 x 25}{1+ \alpha_0 x 55}\]

100+ 5500 α₀ = 120 + 3000 α₀

(5500 – 3000) α₀ = 120-100

2500 α₀ = 20

\[ \alpha_0 = \frac{20}{2500}\]

α₀ = 0.08 per °C

Result

Temperature co-efficient of material α₀ = 0.08 per °C

• The resistance of a material changes when its temperature changes.
• If resistance increases when temperature increases, and vice versa, the material is called a positive temperature coefficient material.
• If resistance decreases when temperature increases, and vice versa, the material is called a negative temperature coefficient material.
• The temperature coefficient of a material also changes with changes in temperature.