Temperature Coefficient of Resistance (Formula And Examples)

As explained in resistance variation with the temperature, the electrical resistance of a material can change when its temperature changes. The size and direction of that change depend on the material and the temperature range.

What is the Temperature Coefficient of Resistance?

Consider a conductor with resistance R₀ at 0^oC and resistance R_t at t^oC.
The linear resistance-temperature relation gives

Here, α_o is the temperature coefficient of resistance referenced to 0^oC.
Within the range where this linear approximation is valid, the change in electrical resistance depends mainly on three factors:

1. the resistance at the initial temperature,
2. the change in temperature, and
3. the temperature coefficient of resistance, α_o.

The value of α_o differs among materials, so equal temperature changes do not produce equal resistance changes in every material.

Under the linear model, the temperature coefficient of resistance at 0^oC is the reciprocal of the material’s extrapolated zero-resistance temperature.

Some materials increase in resistance as they get hotter, while others have an electrical resistance that decreases as temperature rises.

In a metal, higher temperature increases vibration in the crystal lattice. Conduction electrons then undergo more frequent scattering.

This additional scattering impedes current flow, so most pure metals have a positive temperature coefficient and their resistance rises with temperature.

In many semiconductors, however, heating increases the number of mobile charge carriers.

Thermal energy can excite electrons that were previously bound within the crystal, creating additional mobile electrons and holes.

As temperature rises, more electrons gain enough energy to cross the band gap from the valence band into the conduction band.

The extra charge carriers can reduce resistance as temperature rises. Pure silicon, germanium and carbon therefore have a negative temperature coefficient of resistance near room temperature. Semiconductor behaviour is generally nonlinear and also depends on doping, so one constant coefficient does not describe every temperature range.

If resistance changes only slightly over the operating range, the coefficient is close to zero. Constantan and manganin are alloys made to have very small temperature coefficients.

The coefficient is defined at a reference temperature. Its value can also vary with temperature, material purity and physical condition.

When 0^oC is the reference temperature, the coefficient is written as α_o. In the linear model, it equals the reciprocal of the extrapolated zero-resistance temperature.

A coefficient referenced to another temperature is not generally equal to α_o at 0^oC. The conversion below follows from the same linear resistance-temperature model and should be used only across a range where that model remains suitable.

If the temperature coefficient at t°C is αt, it can be calculated from:

The coefficient at t₂^oC can also be written in terms of its value at t₁^oC:

Review the Concept of Temperature Coefficient of Resistance

The electrical resistance of conductors such as silver, copper, gold and aluminum is affected by the scattering of conduction electrons within the material.

Higher temperature increases lattice vibration and electron scattering. The resistance of most metallic conductors therefore rises as their temperature rises.

Suppose a conductor has resistance R₁ at t₁^oC and resistance R₂ at t₂^oC.

For a positive coefficient, the resistance increase (R₂ – R₁) caused by the temperature increase (t₂ – t₁) depends on the following quantities:

Combining these terms gives:

Here, α is the material’s temperature coefficient of resistance at the reference temperature t₁^oC.

Rearranging Equation (1) gives:

If the resistance and temperature coefficient of resistance are known at a reference temperature, Equation (2) estimates the resistance at another temperature within the material’s linear range.

The Temperature Coefficient of Resistance of some Materials or Substances

Representative temperature coefficient of resistance values near 20^oC are listed below. Exact values can vary with purity, composition, physical condition and temperature range.

Sl. No.	Material/Substances	Chemical Symbol/Chemical composition	Temperature coefficient of resistance /oC (at 20oC)
1	Silver	Ag	0.0038
2	Copper	Cu	0.00386
3	Gold	Au	0.0034
4	Aluminum	Al	0.00429
5	Tungsten	W	0.0045
6	Iron	Fe	0.00651
7	Platinum	Pt	0.003927
8	Manganin	Cu = 84% + Mn = 12% + Ni = 4%	0.000002
9	Mercury	Hg	0.0009
10	Nichrome	Ni = 60% + Cr = 15% + Fe = 25%	0.0004
11	Constantan	Cu = 55% + Ni = 45%	0.00003
12	Carbon	C	– 0.0005
13	Germanium	Ge	– 0.05
14	Silicon	Si	– 0.07
15	Brass	Cu = 50 – 65% + Zn = 50 – 35%	0.0015
16	Nickel	Ni	0.00641
17	Tin	Sn	0.0042
18	Zinc	Zn	0.0037
19	Manganese	Mn	0.00001
20	Tantalum	Ta	0.0033

Effect of Temperature on Temperature Coefficient of Resistance of a Material

A material’s temperature coefficient of resistance can change with temperature.

If α_o is the material’s temperature coefficient at 0^oC, Equation (2) gives its resistance at t^oC:

Here, R₀ is the material’s resistance at 0^oC.

Similarly, if the material’s temperature coefficient at t^oC is αt, Equation (2) gives its resistance at 0^oC:

Here, R_t is the material’s resistance at t^o C.

Combining Equations (3) and (4) gives:

Here, α₁ and α₂ are the material’s temperature coefficient of resistance values at t₁^oC and t₂^oC, respectively.

Equation (6) converts a known temperature coefficient of resistance at one reference temperature to another reference temperature within the assumed linear range.

Most pure conducting metals have a positive temperature coefficient of resistance. Their resistance therefore rises as temperature rises.

Many semiconductors have a negative temperature coefficient over a specified range, so their resistance falls as temperature rises. The value and its temperature range depend strongly on the material and its doping.

Alloys such as manganin and constantan have a very small temperature coefficient of resistance.

Their resistance changes only slightly as temperature changes.

This stability makes these alloys useful in precision resistors and measuring instruments.