Trying to understand the temperature coefficient of resistance

[StudentEngine]

Hi,

I am trying to get my head around the temperature coefficient of resistance; that is, the fact that the resistance of a metal (in this case, copper) will change as the temperature of the metal changes. This coefficient is given a value (for copper this is usually given as around 0.00393 per °C) meaning that a metal of resistance 1Ω at 20°C will be at 1.00393Ω at 21°C.

I hope my understanding so far is correct, but what I don't get is that apparently the coefficient itself changes with temperature. So once we get to 30°C or 50°C the rate of change of resistance has changed. Is this correct? I haven't been able to find any sources which give values for this, only for the coefficient at 20°C or 25°C.

Any clarification on this point would be very helpful. Thanks.

Jonathan

 

[Mapes]

Hi StudentEngine, welcome to PF!

Your understanding is correct; Nature has not made it easy and given us a perfectly linear relationship between resistivity and temperature. In contrast, the coefficient itself does change with temperature in general, though it can be approximated as constant along a sufficiently small range. Just think of resistivity as being a general function of temperature:

$$\rho(T)=f(T)\approx\rho_0[1+\alpha(T-T_0)]$$

 

[gsal]

Let's go ahead and start by assigning a letter to this temperature coefficient of resistance; let's use alpha, α.

Now, let it be known that if you have a long piece of copper and you measure its resistance at various temperatures and plot such R.vs.T relationship...you will pretty much get a straight line. In other words, the slope of the line, the rate of change of R per a change in T is constant.

The temperature coefficient of resistance, α, is NOT the slope of such line. It is related to the line, but is not the slope; you can later figure out its relation. For now, let's think of alpha as a constant the permits us to calculate the lump resistance of some copper device at some temperature, based on its known resistance at some other temperature:

R_T2 = R_T1(1+α_T1(T2-T1))

In words, if I know the resistance of my device at temperature T1, I can calculate its resistance at T2, using alpha at T1 and the difference in temperatures, as shown above.

In other words, alpha DOES depend on which temperature we starting from (the temperature the known resistance was measure at) to do our temperature compensation.

For copper, alpha
at T=0°C, α₀ = 0.004264
at T=20°C, α₂₀ = 0.00393

In fact, we can calculate alpha at any temperature using a known alpha at other temperature, let's start from zero, for this example:

α_T= 1 / ( 1/α₀ + T)

Go ahead and practice calculating various resistances by starting at other temperature, etc...

gotta go for now...if this was not enough, ask again.