Copper Resistance: One Model, Not Two Competing Laws

Two copper winding-resistance calculations appear to give different corrected values for the same temperature change. Follow the data path: measure R1 at conductor temperature Theta1, apply a copper temperature model, and calculate R2 at Theta2. The two equations describe the same first-order model only when the temperature coefficient is referenced correctly.

Where does the correction path begin?

Layer one first. For an unchanged conductor, resistance follows the material resistivity:

R = rho × l / A

Here, rho is resistivity, l is conductor length, and A is cross-sectional area. When the same winding is compared at two temperatures, l/A cancels. The calculation therefore corrects the change in resistivity rather than a change in winding geometry.

The usable data path is:

1. Measure winding resistance R1.
2. Associate that measurement with conductor temperature Theta1, in degrees Celsius.
3. Select a copper model whose reference temperature and coefficient are known.
4. Calculate the resistance ratio R2/R1.
5. Multiply R1 by that ratio to obtain R2 at Theta2.

A wrong temperature, mismatched coefficient, or altered conductor path stops the calculation before the equation can produce a meaningful result. A surface temperature that does not represent the winding temperature creates the same problem.

Are the two formulas competing models?

The constant form given for copper is:

R2 / R1 = (Theta2 + K) / (Theta1 + K)

with K = 235 degrees Celsius in the stated simplified form. A more precise value cited for the constant is 234.5 degrees Celsius.

The coefficient form is:

R2 / R1 = 1 + alpha_Theta1 × (Theta2 - Theta1)

Algebra shows that these are identical when the coefficient is referenced to the starting temperature:

alpha_Theta1 = 1 / (Theta1 + K)

1 + alpha_Theta1 × (Theta2 - Theta1)
= 1 + (Theta2 - Theta1) / (Theta1 + K)
= (Theta2 + K) / (Theta1 + K)

The constant equation is not nonlinear in Theta2 when Theta1 is fixed; it produces a straight-line resistance correction. Its quotient form accounts for a starting temperature that may differ from the coefficient's conventional reference point.

Which alpha belongs in the calculation?

The symbol alpha is incomplete unless its reference temperature is known. For the K = 235 model at 20 degrees Celsius:

alpha_20 = 1 / (235 + 20)
         = 1 / 255
         = 0.00392157 per degree Celsius

For K = 234.5 at 20 degrees Celsius:

alpha_20 = 1 / (234.5 + 20)
         = 1 / 254.5
         = 0.00392927 per degree Celsius

The proposed calculation 1/(235+20) = 0.003937 contains an arithmetic error. The correct result using those inputs is 0.00392157 per degree Celsius. A value of 0.003937 therefore must not be paired with K = 235 as though it were derived from that constant.

The supplied material data also distinguish forms of copper at 20 degrees Celsius:

Those values show why a coefficient must travel with its material condition and reference temperature. Select one internally consistent dataset; do not take resistivity from one row, a coefficient from another, and a constant from a third.

Which approach should be used for winding correction?

Use the temperature-constant form when a transformer or motor procedure supplies K = 235 for copper. It directly converts a resistance measured at any stated Theta1 to the required Theta2 without separately translating a 20-degree coefficient.

Use the coefficient form when the conductor documentation supplies alpha and states its reference temperature. If alpha is referenced to 20 degrees Celsius and Theta1 = 20 degrees Celsius, the abbreviated equation is valid:

R2 / R1 = 1 + alpha_20 × (Theta2 - 20)

If Theta1 is not 20 degrees Celsius, retain both reference corrections:

R2 / R1 = [1 + alpha_20 × (Theta2 - 20)]
          / [1 + alpha_20 × (Theta1 - 20)]

For acceptance records or comparisons between tests, use the exact convention named by the applicable equipment procedure. Changing from 235 to 234.5 changes the numerical result, even though both represent the same type of first-order correction.

How is a resistance corrected from 20 to 40 degrees Celsius?

Take Theta1 = 20 degrees Celsius and Theta2 = 40 degrees Celsius. With the simplified constant K = 235:

R2 / R1 = (40 + 235) / (20 + 235)
        = 275 / 255
        = 1.078431

The corrected resistance is therefore:

R2 = 1.078431 × R1

Now derive the matching coefficient at the same starting temperature:

alpha_20 = 1 / (20 + 235)
         = 0.00392157 per degree Celsius

R2 / R1 = 1 + 0.00392157 × (40 - 20)
        = 1.078431

The two results agree because the coefficient was derived from the same K and referenced to Theta1. For comparison, the cited hard-copper coefficient 0.00392 K^-1 produces 1.0784, while the cited soft-copper coefficient 0.0040 K^-1 produces 1.0800. These small differences are material-data and rounding differences, not proof of a separate resistance law.

Why do corrected results disagree?

The first-order equations apply over the temperature range for which the selected material model is intended. The stated resistivity relation was given for -50 to +200 degrees Celsius. Extrapolating the K = 235 expression to -235 degrees Celsius would predict zero resistance even though that temperature remains above absolute zero, demonstrating that K is a fitted temperature constant rather than a universal physical endpoint.

How should the result be calculated and verified?

1. Record R1, Theta1, the target Theta2, conductor material, and the governing test convention.
2. Confirm that both temperatures are in degrees Celsius. Temperature differences have the same numerical magnitude in kelvins, but the additive constant form requires Celsius values.
3. Choose one model: the supplied K form or a documented coefficient with its reference temperature.
4. For the recommended winding-correction path, calculate F = (Theta2+K)/(Theta1+K).
5. Calculate R2 = R1 × F without rounding intermediate values prematurely.
6. Perform an independent check by deriving alpha_Theta1 = 1/(Theta1+K) and calculating Fcheck = 1+alpha_Theta1(Theta2-Theta1).
7. Confirm that F and Fcheck agree to the retained precision and that resistance rises when copper temperature rises.

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