Electronics > Metrology

Resistivity vs Temperature – flatter is better?

**Zeranin**:

This is a sister thread to my ‘PCR vs TCR’ thread. We need to agree on the fundamentals, so here I will talk in general about resistance and resistivity. We all know what resistivity is. Resistivity is a property of the material, but not of the dimensions of the material. Resistance depends upon resistivity, and the dimensions, as per :-

R = rho x L / A

Where rho is the resistivity of the material in ohm-meter

L is the length of material, in meters

A is the cross sectional area of the material, in meters^2

R is the resistance, in ohms

I need to briefly mention Temperature Coefficient of Resistance (TCR), typically expressed in units of ppm/K. Frankly, most of the common ways that TCR is expressed are pretty useless and often deliberately misleading, but I think we would all agree that what is fundamentally useful is the plot of Resistance versus Temperature, from which the TCR is always determined anyway. As far as I am concerned, if the term TCR means anything at all, then it is the slope of the R-T curve, which of course should be as small as possible. In my view, its best to leave the vague term ‘TCR’ right out of discussions, and instead talk about the R-T curve, and the slope of that curve, dR/dT. If you know the R-T curve, then you know everything there is to know, whereas ‘TCR’ can mean almost anything and frequently nothing.

For many years I assumed without thinking about it that the perfect resistive material would have a resistivity that was rock constant with respect to temperature, yet if you think about it, that is wrong. We don’t actually care if the resistivity is constant with temperature, what we actually want is for the resistance to be constant with temperature. The resistance depends not only on the resistivity, but also on the length and cross sectional area of the material from which the resistor is constructed.

Most materials (including resistive alloys) expand and contract with temperature. For example, Zeranin has a thermal coefficient of expansion (COE) of 18ppm/K. Therefore, if we manufacture a resistor from a round bar of Zeranin, and change the temperature by 1 DegC, then the length will increase by 18ppm, while the diameter will also increase by 18ppm. As the cross sectional area scales as the square of the diameter, this means that the cross sectional area will increase by 36ppm. You can see where this is leading. If the length increases by 18ppm, and the cross sectional area increases by 36ppm, then the net effect is to decrease the resistance by 18ppm/K, quite independently of any change of resistivity with temperature.

It follows that if we could concoct a magic alloy whose resistivity did not change with temperature, but had a COE of +18ppm/K, as does Zeranin, then resistors built from this magic alloy would be rubbish, with a linear change in resistance with temperature of -18 ppm/K. Clearly, the name of the game is not to produce alloys whose resistivity does not change with temperature, at least not unless the alloy also has a zero COE, which is rarely the case.

What the resistor alloy alchemist actually does, is to empirically produce an alloy with the flattest resistance-temperature curve. In doing so, what he is actually doing, though may not realize it, is matching the Resistivity-Temperature curve to be equal and opposite to the linear change in resistance with temperature that comes about from the coefficient of expansion. For example, in the case of Zeranin, at the temperature where dR/dT=0, the resistivity of the material must change by +18ppm/k, exactly cancelling the -18ppm/K change in resistance due to thermal expansion. I have not seen this spelled out in any textbook, yet evidently this must be true. The best resistors, with very flat R-T curve, have a resistivity that varies significantly with temperature, and the resistance also varies considerably with temperature on account of the COE, but the two effects have been arrange to cancel. This concept will be important in my discussion on the PCR vs TCR thread.

Does anyone disagree with any of this? If no one shoots this posting down in flames, then I’ll continue my discussion on the sister thread, PCR vs TCR.

**Vgkid**:

This could be another interesting read. To bad Edwin hasn't been online in almost 2 weeks. You could try sending him a pm/email.

**uncle_bob**:

Hi

Consider that most modern resistors are *not* made of wire hanging loosely in free space. Once you put a core material under them, or wrap them on a form, you add another variable.

One example:

Take your magic material and thin film deposit it on a glass substrate. The glass now dominates the X and Y dimensions of the device. The Z is still dependent on the magic material.

Yes, it's complicated.

Bob

**Zeranin**:

--- Quote from: uncle_bob on April 22, 2016, 12:25:03 am ---Hi

Consider that most modern resistors are *not* made of wire hanging loosely in free space. Once you put a core material under them, or wrap them on a form, you add another variable.

One example:

Take your magic material and thin film deposit it on a glass substrate. The glass now dominates the X and Y dimensions of the device. The Z is still dependent on the magic material.

Yes, it's complicated.

Bob

--- End quote ---

All true. It's actually fairly easy to analyse the case of a wire-wound or thin film 'winding' over a cyclindrical substrate. Assuming that the R-T curve of the naked material is parabolic, as it usually is, then the result of winding on a cylinder with different coeffcient of expansion is simply to add an additional linear term in the R-T curve, that has the effect of shifting the temperature of the turning point, which can be even be useful, and resistor manufacturers therefore sometime deliberately choose to have different COE for the wire and former. I don't want to go off on a tangent though, I'll happily go through the theory on another thread some time if you like. This thread refers to a 'naked' resistor.

**uncle_bob**:

Hi

The main point is that just as you can't make a bulk part that does not change dimension with temperature, other very similar things quickly get into the mix as well. If you are going to consider one, it's useful to recognize that it's only a part of the whole design problem. The interaction with the "next guy in line" may very well impact you analysis of a free hanging wire.

Bob

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