# 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,...)? – stevenvh Sep 28 '12 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? – Jason S Sep 28 '12 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. – Jason S Sep 28 '12 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! – Jason S Sep 29 '12 at 0:26
• So this applies to your wiring if that wiring is pure copper. – Kaz Sep 29 '12 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 – Jason S Sep 29 '12 at 13:47
• en.wikipedia.org/wiki/… : Today, copper conductors used in building wire often exceed the 100% IACS standard. – Jason S Sep 29 '12 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%. – Jason S Oct 1 '12 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.

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. – Jason S Sep 28 '12 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[0],r[0],4), np.polynomial.chebyshev.Chebyshev.fit(T[1],r[1],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 http://library.bldrdoc.gov/docs/nbshb100.pdf

• 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 '12 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. – Jason S Oct 3 '12 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 '12 at 8:55