RTD linearization

Exponential? How so?

John

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Thranx. I learned.

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Each degree of temperature change is a multiplier on the degree base = before it. so the recurrence relation results in an exponential. Thus it can be = modeled=20 as r' =3D r(25) * k(1)e^[k(2)*t], an exponential. BTW just look at a R = vs T plot.

One interesting thing is that the self-heating of this 1206 part is so low if you solder it to a lot of copper. We've verified that with infrared measurement of the hot-spot temp on conventional thick-film resistors. Since most surface-mount resistors are the same thickness of alumina, theta from the element to their end caps depends only on their l/w ratio, which also tends to be the same. So an 0603 resistor can handle as much power as a 1206, half a watt maybe, if you mount it right.

Some people are making resistors on AlN substrates, and they have hard-to-believe power ratings.

John

What are the coefficients? The curve slopes down.

I've never seen the platinum RTD curve expressed as an exponential. The usual formulation is a polynomial.

John

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A curve whose normalized slope is constant is an exponential--for instance, tempco of resistance is generally defined as (1/R)dR/dT, which is d/dT(ln R). If that were really some constant alpha, then ln R would be proportional to T, so R would be something times exp(alpha T).

Of course, there's nothing that says the tempco is constant, IOW not all curves are exponential. The RTD curve is a simple rational function of T--it really linearizes beautifully with a bit of negative resistance, with theoretical deviations are of the order of 0.01K over wide temperature ranges.

Cheers

Phil Hobbs

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Maybe 0.1°C over the full temperature range of an RTD (-200 ~ 850°C). R = -2311 ohms.

About +/-0.02° for 0 ~ 400°C. R = -2515 ohms

And almost nothing for 0 ~ 100°C R = -2687 ohms

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I can see we share some of the same odd taste in recreations.

Depends a bit on whether you like the European or US curves, of course. They have shinier platinum over there, like everything else--just ask them. ;) But then at that level it also depends on residual stress in the metal, CTE mismatch with the substrate....

Still, from a mathematical perspective, it's pretty.

Cheers

Phil Hobbs

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Interesting, i listen to you on this but not to JL.

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The "R" is calculated using a simple equation to exactly zero the linearity error out at mid-scale**. IOW, so Rt(Tmid) - Rt(Tzero)= 0.5* (Rt(Tfs) - Rt(Tzero)), where Rt is the parallel combination of the linearization resistor and the sensor Rt = Rsensor(T)*R/(Rsensor(T) + R)

This method of linearization is almost perfect over a relatively narrow span, and pretty good even over the whole usable range of the sensor (of course 100°C -- say -40°C ~ +60°C is a very wide range for human comfort, but only about 10% of the useful range of a general-purpose sensor).

** Since it's an imperfect fit, you have to come up with some kind of criteria for judging what is the 'best' fit. For ranges of a few hundred C or less, it looks like zeroing the error at midscale is pretty reasonable-- an S-shaped error curve that is centered on zero error and crosses the zero line at zero, full-scale and mid scale.

OTOH, over the full range, it yields asymmetrical peak errors so it may not be optimal. Sometimes cost functions can be pretty strange- mathematicians like to use sum (or integral) of error squared, because it's easy to differentiate and find a closed form solution for the minimum error-- in some cases. Instrument designers, OTOH, are aiming for specs like worst-case error PLUS maybe there should be no systemic error at calibration points and/or some easy-to-check point like room temperature (or body temperature) for a temperature instrument or ~21% for an oxygen meter. Nobody in the real world is likely to request a sensor or instrument with a maximum integral of error squared over some range. ;-)

Anyway, these days we can often use a processor or FPGA to apply simple brute-force numerical techniques and reduce linearization errors to negligible levels in relation to the error budget-- or even below the noise floor. Thermocouples have ugly (but fairly small) nonlinearity... nothing a set of double-precision polynomials curve-fitted to the values can't handle. The same method can be used to linearize RTDs when you have a universal-input instrument-- no need to be clever for that part of the design... in fact it could be counterproductive.

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You still haven't put numbers on your exponential RTD thing.

John

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OK. I see now, it works much like the changes in quartz crystal cuts in=20 the 1960s through 1980s, with doubly rotated AT (DRAT) and later SC cuts. Set things up to create a "flat spot/range" in the tempco curve shape.

Thermal expansion coefficient (1/L)dL/dT, is similarly defined.

--
"Electricity is of two kinds, positive and negative. The difference
is, I presume, that one comes a little more expensive, but is more
durable; the other is a cheaper thing, but the moths get into it."
                                             (Stephen Leacock)