# reference source for copper conductivity vs. temperature?

I am looking for a curve for copper conductivity vs. temperature over the -55°C to +180°C range (covers std. ranges of semiconductors plus NEMA class H insulation in the case of magnetics / motors) but cannot seem to find anything searching on the internet.

I have heard that it depends on the particular copper alloy as well, I'm just looking for "normal" copper found in electrical wire.

Does anyone have a reference that would include this information?

edit: I'm not looking for a linear approximation (the Wikipedia page on conductivity lists copper conductivity and its temperature coefficient at 20°C) but rather a curve.

• I don't know about copper, but many (most?) metals will have a near linear curve. What makes that the linear approximation isn't precise enough? Especially since you talk about "cable", without even naming the production process (annealing, quenching,...)? Sep 28, 2012 at 18:58
• linear in resistance? is it proportional to abs. temperature? meaning the conductivity would be inversely proportional to abs. temperature? What reference source can I confirm that from? Sep 28, 2012 at 19:00
• ...and in my view, you cannot use a linear approximation unless you know what range it is meant to apply to, and what kind of errors you can expect over that range from this approximation. Sep 28, 2012 at 19:04

Data from NIST (table 2)

At low temperatures the resistivity of copper approaches a "residual" resistivity. However, this isn't really a concern until below ~100K.

At higher temperatures the resistivity is best approximated with a linear increase vs. temperature (as you've already found). I plotted the resitivity vs. temperature, and I wouldn't be too worried about non-linearity too much until ~800K. • ooh! empirical data + some theory behind it in the NIST doc. Thank you! Sep 29, 2012 at 0:26
• So this applies to your wiring if that wiring is pure copper.
– Kaz
Sep 29, 2012 at 2:49
• No, table 2 contains data for annealed copper 99.999% pure or greater; this corresponds to the IACS standard ndt-ed.org/GeneralResources/IACS/IACS.htm which is what wire manufacturers should be referencing in their specs Sep 29, 2012 at 13:47
• en.wikipedia.org/wiki/… : Today, copper conductors used in building wire often exceed the 100% IACS standard. Sep 29, 2012 at 13:50
• I'm adding a followup answer to discuss the NIST table 2 data more quantitatively; the conclusion "I wouldn't be too worried about non-linearity" over a certain range can't be made visually; the eyes are bad at judging nonlinearities less than about 5-10%. Oct 1, 2012 at 13:47

The Bloch-Grüneisen Formula gives you a function of resistivity vs. temperature.

Debye temperature Θ is a parameter of the function.
Its value for pure Copper is ΘCu = 343.5 K

If you can't find the value of the Debye temperature Θalloy of your particular alloy, you can make some measuremnts over the interesting temperature range and determine it by least-square fitting.

• Is there a closed-form expression (or at least analytic form) of the above? Performing an integral for each function evaluation is not great.
– TLW
May 14, 2022 at 20:40
• I guess if there was a known closed-form expression the final formula would not have been published as integral. Googling, however, finds at least this: An accurate analytic representation for the Bloch-Gruneisen integral
– Curd
May 15, 2022 at 10:36

There are lots of resources out on the interwebs for information about copper; they even have their own website.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/restmp.html

• thanks, but I'd already found copper.org; it doesn't seem to have a copper vs. temperature curve. The equation you gave me is a linearization (which I can get from Wikipedia); I'm looking for a more detailed curve. Sep 28, 2012 at 18:49

To follow up on helloworld922's response, here is some quantitative analysis of the NIST Table 2 data, both from 100-800 K and from 200-500K (=-73.15K to +226.85 K, a range that is closer to the working range of motor windings) using PyLab:

# copper conductivity fit from NIST:
# Journal of Physical and Chemical Reference Data
# Electrical Resistivity of Copper, Gold, Palladium, and Silver
# R. A. Matula
# JPCRD 8(4) pp. 1147-1298 (1979)
# http://www.nist.gov/data/PDFfiles/jpcrd155.pdf
# numerical data from table 2

import numpy as np
import matplotlib.pyplot as plt

T0 = np.array([ 100, 125, 150, 175, 200, 225, 250, 273.15, 293, 300, 350, 400, 500, 600, 700, 800])
r0 = np.array([ 0.348, 0.522, 0.699, 0.874, 1.046, 1.217, 1.387, 1.543, 1.678, 1.725, 2.063, 2.402, 3.090, 3.792, 4.514, 5.262])
T = (T0, T0[4:-3])
r = (r0, r0[4:-3])
f = [np.polynomial.chebyshev.Chebyshev.fit(T,r,4), np.polynomial.chebyshev.Chebyshev.fit(T,r,4)]

for i in range(0,2):
plt.figure(i+1)
plt.plot(T[i],r[i]-f[i](T[i]))
plt.plot(T[i],r[i]-f[i].truncate(2)(T[i]))


What this does is to approximate the resistivity vs. temperature data with 4th-degree polynomials, and plot graphs of errors in the polynomial approximation vs. temperature, for the 4th-degree polynomial and for a linear polynomial.

The Chebyshev coefficients* for a polynomial fit are:

100-800K: 2.77689151e+00, 2.44286730e+00, 3.00859915e-02, 1.44364826e-02, -2.64166679e-03

200-500K: 2.06557521e+00, 1.02015113e+00, 2.57505898e-03, 1.80482498e-03, -1.11397377e-04

Over 100K - 800K: Here I would definitely say that a linear approximation is close but not sufficient for precision applications. The error in a linear approximation is close to 1% of the resistivity value (slightly higher for <200K). I would use a quadratic or cubic approximation. (the 4th degree coefficient is small enough that it's not worth the extra complexity)

Over 200K - 500K: Here the linear approximation does much better due to the reduced range; maximum error of a linear approximation is about 0.15% of the resistivity value. I'd probably live with a linear approximation.

*Chebyshev coefficients: Chebyshev approximation is a technique for approximating functions with polynomials + quantitatively evaluating the accuracy of those approximations. The subject is too detailed to discuss it much here. If you have a copy of Numerical Recipes go read about Chebyshev approximation. I also just wrote a blog post about Chebyshev approximation which might help explain.

edit: The JPCRD Matula reference refers to ultrapure copper exceeding IACS levels. "Standard" commercial copper is slightly lower conductivity than ultrapure (about 2.5% lower) and has a lower tempco of 0.00393. See NBS Handbook 100

• Why do you want to use Chebyshev polynomals for a approximation if a resistivity-vs-termeperature formula is known (Block-Grüneisen-Formula)? Using that formula for a least square fit (e.g. with python using 'leastsq' from module 'scipy.optimize') will get you only two parameters (C and &Theta;) insead of a whole bunch of Chebyshev coefficients that have no physical meaning and probably serve the purpose less accurate.
– Curd
Oct 2, 2012 at 21:41
• I don't care about an exact approximation. For that matter, if I can get away with a linear equation, I don't care about Chebyshev at all. What I care about is how close the result is to linear. Chebyshev coefficients tell me that. Oct 3, 2012 at 0:28
• What you have done doesn't look like you didn't care about Chebyshev polynomals :-) . Just for the purpose of checking how close a linear approximation is you could have done a simple linear regression and look at the residuals.
– Curd
Oct 3, 2012 at 8:55