In this article, we will learn about the Temperature Coefficient of Resistance, which is the most important parameter that describes the change in the resistance of a material with a variation in its temperature. Thus, it is defined as the measure of variation in the electrical resistance of a material with change in temperature. From the basic theory of the Effect of Temperature on Resistance, we know that the resistance of metals increases with an increase in temperature, and the resistance of insulators and semiconductors decreases with an increase in temperature. Hence, the temperature coefficient of resistance explains the behavior of resistance variation with temperature change, and its study plays an important role in selecting the right material to make electrical conductors and insulators. This chapter will provide a comprehensive study of the temperature coefficient of resistance, including its definition, unit, formula, types, and solved examples.
What is Temperature Coefficient of Resistance (TCR)?
The fractional variation in electrical resistance with the per-unit change in temperature is called the temperature coefficient of resistance (TCR). It is usually denoted by the symbol α. The temperature coefficient of resistance provides a measure of the variation in resistance of a material with change in temperature.
Relation between Temperature and Resistance
Let us now derive the relation between temperature and resistance. For this, consider a conductor having an electrical resistance of R0 ohms at 0 °C and Rt ohms at t °C. Then, in the normal range of temperature, the change in resistance is given by,
$$ΔR=R_t-R_0$$
From experiments, it is seen that the change in resistance ΔR –
- Is directly proportional to the initial resistance
- Is directly proportional to the change in temperature
- Depends upon the nature of the material
From the first two points,
$$(R_t-R_0 )∝R_0$$
And
$$(R_t-R_0 )∝(t-0)∝t$$
Hence,
$$(R_t-R_0 )∝R_0 t$$
$$⇒(R_t-R_0 )=α_0 R_0 t\;\;\;…(1)$$
Where α0 is a proportionality constant and is called as temperature coefficient of resistance at 0 °C.
On rearranging equation (1), we get,
$$R_t=R_0 (1+α_0 t)\;\;\;…(2)$$
From equation (2), we can derive the formula of the temperature coefficient of resistance as,
$$α_0=\frac{(R_t-R_0)}{R_0 t}\;\;\;…(3)$$
Temperature Coefficient of Resistance at Various Temperatures
Suppose α0, α1, and α2 are temperature coefficients of resistance at temperatures t0, t1, and t2, respectively, then temperature coefficients of resistance at these temperatures can be given as follows –
$$α_1=\frac{α_0}{(1+α_0 t_1)}\;\;\;…(4)$$
$$α_2=\frac{α_0}{(1+α_0 t_2)}\;\;\;…(5)$$
This can be determined for any temperature in the same way.
Special Case of Resistance Temperature Relationship
Let us consider a conductor having resistance R1 at t1 °C and R2 at t2 °C and α1 is the temperature coefficient of resistance at t1 °C, the temperature-resistance relation can be written as,
$$R_2=R_1[1+α_1(t_2-t_1)]\;\;\;…(6)$$
$$R_2=R_1 (1+α_1 Δt)\;\;\;…(7)$$
Temperature Coefficient of Resistance Formula
From the above discussion, we can write a standard expression or formula to calculate the temperature coefficient of resistance for a given temperature difference and resistance variation as,
$$α=\frac{1}{R_0}×\left(\frac{ΔR}{ΔT}\right)\;\;\;…(8)$$
Here,
- R0 is the initial resistance, usually taken at 0 °C or 20 °C.
- ΔR is the change in resistance.
- ΔT is the change in temperature.
Here, it is also important to note that the relations derived above are the linear approximations and work over a moderate range of temperature. At very low or high temperatures, higher-order corrections are required.
Unit of Temperature Coefficient of Resistance
From equation (8), we can see,
$$\text{Unit of }α=\frac{1}{Ω}×\frac{Ω}{°C}=\frac{1}{°C}$$
Thus, the unit of temperature coefficient of resistance is per degree Celsius.
Types of Temperature Coefficient of Resistance
The temperature coefficient of resistance can be classified into the following two types –
- Positive Temperature Coefficient of Resistance – When the resistance of a material increases with the increase in temperature, then the temperature coefficient of the material is called positive TCR.
- Negative Temperature Coefficient of Resistance – When the resistance of a material decreases with the increase in temperature, then the temperature coefficient of the material is called negative TCR.
The following figure graphically illustrates the positive and negative temperature coefficients of resistance.
Values of Temperature Coefficient of Resistance
Depending on the atomic structure and electron mobility, different materials have different values of the temperature coefficient of resistance. The following table gives typical values of the temperature coefficient of resistance for different materials –
|
Material |
Temperature Coefficient of Resistance (per °C at 20°C) |
| Silicon (Si) |
–0.07 |
| Mercury (Hg) |
0.0009 |
| Brass |
0.0015 |
| Gold (Au) |
0.0034 |
| Silver (Ag) |
0.0038 |
| Copper (Cu) |
0.00386 |
| Platinum (Pt) |
0.003927 |
| Tin (Sn) |
0.0042 |
| Aluminum (Al) |
0.00429 |
| Tungsten (W) |
0.0045 |
| Nickel (Ni) |
0.00641 |
| Iron (Fe) |
0.00651 |
Numerical Problems on Temperature Coefficient of Resistance
The following are some solved numerical problems included here to explain the application of formulas and concepts explained above.
Q. 1 – The electrical resistance of a copper wire at 0 °C is 10 Ω. If the temperature coefficient of resistance of copper is 0.00386 /°C, then find the resistance of the wire at 50°C.
Solution – Given,
$$R_0=10\;Ω$$
$$α=0.00386\;°C^{-1}$$
Then, the resistance of the given copper wire at 50 °C will be,
$$R_{50}=R_0 [1+α(50-0)]$$
$$R_{50}=10×[1+(0.00386×50)]$$
$$R_{50}=11.93\;Ω$$
Q. 2 – An iron wire has a resistance of 12 Ω at 0 °C, and its resistance at 100°C is 19.8 Ω. Determine the temperature coefficient of resistance of iron wire.
Solution – Given
$$R_0=12\;Ω$$
$$R_{100}=19.2Ω$$
Then, the temperature coefficient of resistance of iron will be
$$α=\frac{1}{R_0}×\frac{(R_{100}-R_0)}{(100-0)}$$
$$α=\frac{1}{12}×\frac{(19.2-12)}{100}=\frac{7.8}{(12×100)}$$
$$α=0.0065\;°C^{-1}$$
Q. 3 – The resistance of a platinum wire is 5 Ω at 20°C. If the resistance becomes 5.98 Ω at 100 °C, then calculate the temperature coefficient of resistance of platinum wire.
Solution – Given,
$$R_{20}=5\;Ω$$
$$R_{100}=5.98\;Ω$$
The temperature coefficient of resistance of platinum will be,
$$α=\frac{1}{R_{20}}×\frac{(R_{100}-R_{20})}{(100-20)}$$
$$α=\frac{1}{5}×\frac{(5.98-5)}{80}=\frac{0.98}{400}$$
$$α=0.00245\;°C^{-1}$$
Q. 4 – The resistance of a coil is 100 Ω at 20 °C. When the coil’s temperature is increased to 120°C, its resistance becomes 120 Ω. Calculate the temperature coefficient of resistance of the coil.
Solution – Given,
$$R_{20}=100\;Ω$$
$$R_{120}=120\;Ω$$
The temperature coefficient of the coil will be,
$$α=\frac{1}{R_{20}}×\frac{(R_{120}-R_{20})}{(120-20)}$$
$$α=\frac{1}{100}×\frac{(120-100)}{100}$$
$$α=\frac{20}{(100×100)}=0.0002\;°C^{-1}$$
Q. 5 – A wire has an electrical resistance of 15 Ω at 30°C. If the temperature coefficient of resistance is 0.004 /°C, determine the resistance at 150°C.
Solution – Given,
$$R_{30}=15\;Ω$$
$$α=0.004\;°C^{-1}$$
The resistance of the wire at 150 °C will be,
$$R_{150}=R_{30} [1+α(150-30)]$$
$$R_{150}=15×[1+(0.004×120)]$$
$$R_{150}=15×1.48=22.2\;Ω$$
Q. 6 – The resistance of a conductor wire at 0 °C is 50 Ω. At what temperature will the resistance of the wire become 75 Ω, if α=0.005 /°C?
Solution – Given,
$$R_0=50\;Ω$$
$$R_T=75\;Ω$$
$$α=0.005\;°C^{-1}$$
The temperature T at which the resistance will become 75 Ω will be,
$$R_T=R_0 [1+α(T-0)]$$
$$75=50[1+0.005T]$$
$$⇒1+0.005T=\frac{75}{50}$$
$$⇒T=\frac{0.5}{0.005}=100\;°C$$
Q. 7 – A thermistor has a negative temperature coefficient of resistance. Its electrical resistance at 27 °C is 1000 Ω, and its resistance becomes 500 Ω at 57 °C. Find the effective temperature coefficient of resistance in this range of temperature.
Solution – Given,
$$R_{27}=1000\;Ω$$
$$R_{57}=500\;Ω$$
The temperature coefficient of resistance of the thermistor will be,
$$α=\frac{1}{R_{27}}×\frac{(R_{57}-R_{27})}{(57-27)}$$
$$⇒α=\frac{1}{1000}×\frac{(1000-500)}{30}=\frac{-500}{(30×1000)}$$
$$α=-0.0167\;°C^{-1}$$
Conclusion
Hence, this is all about an important parameter related to the electrical resistance of materials, called the Temperature Coefficient of Resistance. TCR plays a critical role in understanding the behavior of the electrical resistance of materials with the change in temperature. Thus, it helps in selecting the right materials for applications.
FAQs About Temperature Coefficient of Resistance
In this section, we have collected some most Frequently Asked Questions (FAQs) related to the temperature coefficient of resistance and tried to briefly answer them.
1 – What is the formula for temperature to resistance?
Ans – The formula of the temperature coefficient of resistance is given below –
$$α=\frac{1}{R_0}×\frac{ΔR}{ΔT}$$
Where,
- R0 is the initial resistance
- ΔR is the change in resistance
- ΔT is the change in temperature
2 – How do you calculate TCR?
Ans – The temperature coefficient of resistance (TCR) can be calculated using the following formula –
$$α=\frac{1}{R_0}×\frac{ΔR}{ΔT}$$
3 – What does the temperature coefficient of a resistor mean?
Ans – The temperature coefficient of a resistor is a factor that indicates the variation in the resistance with per °C change in the temperature of the resistor.
4 – What is the zero temperature coefficient?
Ans – In terms of the temperature coefficient of resistance, Zero Temperature Coefficient is defined as the value of TCR for which the resistance of the material will not change significantly with a change in temperature.
5 – What is the temperature coefficient of resistance of Cu?
Ans – The temperature coefficient of resistance of copper (Cu) at 20 °C is 0.00386 /°C.