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R as a function of temp in NTC material

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kell

Wed Oct 25 2006, 04:35PM

Registered Member #142 Joined: Sat Feb 11 2006, 01:19PM
Location:
Posts: 102

I came into posession of an in-line oil heater sold for heating vegetable oil.
Turns out it has a negative temperature coefficient. I measured the resistance at several temperatures and did some curve-fitting just for giggles.
32 F: R = 2.5 ohm
74 F: R = 1.85 ohm
212F: R = 1 ohm
First I though about quadratic regression, but you don't get asymptotes with a quadratic. The actual resistance of the material would have an asymptote for temperature increasing without limit (somthing less than 1 ohm, but obviously not a negative value).
So I decided to look at a kind of hyperboloid with three coefficients: (R+a)(T+b)=c. After fussing around with the numbers some, I decided to drop the a (making the resistance asymptote zero).
This only leaves two coefficients to fit three data points, but check it out. I solved for b and c using the data points at 74 and 212 degrees; plugged in the third data point, and it fit. Thought whoa, maybe I'm onto something here.
The equation: R(T+88.4)=300
Looking at it for physical significance, obviously R approaches 0 as temperature increases without limit.
But what about that 88.4? The equation predicts as temperature approaches -88.4 F, resistance should increase without limit. No conductor of course has infinite resistance, and you would think that if it gets cold enough (like approaching abs zero), the resistance would start dropping. Below -88.4 F the equation has no physical meaning.
But might be the curve still gives a good prediction of the material's behavior at temperatures higher than -88.4 F. It isn't hard to imagine the material showing, if not infinite resistance, megohms at temperatures around the theoretical -88.4, then tracing the curve as predicted.
I guess I have too much time on my hands.

Edit:
I decided to go ogle the internet.
Found mention of third-degree polynomials and something called the Steinhart-Hart
equation, 1/T=A+B(lnR)+C(lnR)^3 supposed to result in great accuracy when fitted over
50 deg C intervals.
Also read that the R-T characteristic is supposed to be a
"negative, non-linear exponential function."
So I fitted the 74 and 212 degree data points, using Rankine values now into an equation of the form
R = a exp(b/T) and got R = .0929 exp(10.76/T).
That predicts R = 2.39 at 32 degrees. Probably as good a fit as the hyperbolic, considering
measurement error. And a better fit with physical reality, because the hyperboloid
exhibits an anomaly at -88.4 F. Sort of like predicting absolute zero is -88.4 degress F
instead of -459.
So I think the fact that the hyperboloid fit so well is a coincidence of sorts.
But it points to a weakness we humans have in looking for theories to fit the facts we have.
It was such a good fit, I thought it must mean something.

Sulaiman

Wed Oct 25 2006, 07:54PM

Registered Member #162 Joined: Mon Feb 13 2006, 10:25AM
Location: United Kingdom
Posts: 3141

My math is a bit rusty but I believe that
for a linear relationship at least two data points are required
and at least one more data point for each increase in 'order'
so with only three data points you either use a simple quadratic fit
(as you did) or else many possible higher order curves will fit
with no way to know which (if any) is true.
I think that taking many more measurements is required
before you can determine the true curve equation.

Steve Conner

Thu Oct 26 2006, 12:05AM

Registered Member #30 Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706

What kind of dumb ass makes a heater with a negative temperature coefficient?

That's almost as stupid as if they sold a toxic substance in bars that affected your brain when you drank it, making you want to drink even more of it.

FWIW, I use the Steinhart-Hart equation quite often at work for calibrating temperature readouts derived from NTC thermistors.

Carbon_Rod

Thu Oct 26 2006, 02:39AM

Registered Member #65 Joined: Thu Feb 09 2006, 06:43AM
Location:
Posts: 1155

Intro to Cubic Spline Interpolation:

And the simple MatLab plot program in the zip.

Cheers,
:D
]1161830397_65_FT17327_c_spline.zip[/file]

Simon

Thu Oct 26 2006, 03:42AM

Registered Member #32 Joined: Sat Feb 04 2006, 08:58AM
Location: Australia
Posts: 549

For three points you aren't going to get many cubic splines. Splines are only good for interpolation, not extrapolation, anyway.

Fundamentally, you've only got a few data points with a range from 32F to 212F. You can fit all kinds of curves but unless you have some good theoretical grounds for your choice, the results will only really make sense between about 32F and 212F.

(Sulaiman: you aren't so rusty. You're right about the "two points for a line, three for a quadratic, four for a..." thing.)

wrote ...

But it points to a weakness we humans have in looking for theories to fit the facts we have.
It was such a good fit, I thought it must mean something.

That's the lesson of a true scientist!

A key concept here is "degrees of freedom". This is essentially the number of independent numbers you can twiddle. A set of three independent measurements has three degrees of freedom. A quadratic has three completely independent coefficients so you can always perfectly match a quadratic to three data points.

Whenever you have a model that has at least as many degrees of freedom as your data, you can generally tweak things to get a perfect fit.

In your case, your model had one less degree of freedom than your data. While you can't ensure a perfect fit, that doesn't mean a good fit is anything special.

kell

Thu Oct 26 2006, 10:04PM

Registered Member #142 Joined: Sat Feb 11 2006, 01:19PM
Location:
Posts: 102

Conserve Ten wrote ...

What kind of dumb ass makes a heater with a negative temperature coefficient?

Exactly.
I got into a little argument on another board where a lot of the netizens use the aforementioned heater in their diesel vehicles for preheating vegetable oil to run the engine on (as an alternative fuel). I said undiplomatic things like "bad news" and "retarded" so of course that set everybody off.
I'm just going to get some stainless steel wire or nichrome or something and wrap it around a metal tube with fibreglass or asbestos tape. But of course I'll use a material with a positive tempco, which includes all commonplace conductors.
I guess the designers of the "vegtherm" decided they had to go with something really robust, like a ceramic. Which has a negative tempco and probably costs a lot, to boot. Go figure.

Carbon_Rod

Fri Oct 27 2006, 02:11AM

Registered Member #65 Joined: Thu Feb 09 2006, 06:43AM
Location:
Posts: 1155

In a ludicrous kind of way it could make some sense as during the preheat stage it draws less power while the battery is under maximum load from the starter. Also, at optimal temperature the thermostat should switch the power off automatically (or an adjustable thermo-breaker like the one on a home hot-water-heater.)

I thought about trying this with an older diesel truck (metal piping to injectors):
1.) Dual fuel tanks with a automatic fuel line switch.
2.) Tank 1 standard Diesel fuel to start the engine.
3.) Tank 2 experimental fuel through a thick double walled stainless steel pipe wrapped around the exhaust manifold. Adjust diameter for optimal flow rate versus fuel degradation.
4.) Transfer tanks slowly while running hot.

I assumed you were trying to build some sort of PWM control system.

Steve Conner

Fri Oct 27 2006, 09:20AM

Registered Member #30 Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706

Kell, I bet the designers ordered the wrong kind of ceramic by accident

Or maybe they had their hearts set on a ceramic heater, and the only ceramic with low enough resistance to work at 12V was NTC.

I see lots of heaters made out of PTC ceramic material, which makes for very stable and safe behaviour, pretty much like a built-in thermostat. But I've never ever heard of one made of NTC, until now.

FWIW, at 12V, you could maybe even get away with passing current through the vegetable oil line itself. If you chose a small thin-walled steel pipe, it might not be too heavy a load. There's probably some really good reason I never thought of as to why people don't do this, though.

kell

Fri Oct 27 2006, 04:22PM

Registered Member #142 Joined: Sat Feb 11 2006, 01:19PM
Location:
Posts: 102

I'd have a look at steel's bulk resistivity and figure out how long and thin the pipe I'd need to see if that's practical.
On the subject of NTC heating, I thought up a little circuit to keep the temperature of the heating element confined in a window.
It uses an op-amp (or a comparator, with the addition of a pull-up resistor) and resistors in a bridge configuration is such a way that it regulates current and keeps the resistance of the NTC conductor constant, which will keep the temp constant also.
If the heating element ever gets up to heat (!), its resistance should cycle in a narrow range about the value
Rs(R2/R1). The feedback resistor on the positive input of the op-amp should have a much higher value than the resistors in the divider.

V in
                  |
                  |
  ,---------------+---------------------,
  |                                     |
  |                                     |
  /                                     /
  \                                     \NTC
  /R2                                   /heater
  \                                     \
  /                                     /
  |                                     |
  |                                     |
  |     ,--/\/\/\/---,                  |
  |     |            |                  |
  |     |  |\        |                  |
  +-----+--|+\       |     10        |--'
  |        |  >------+--/\/\/\/--+--||
  |    ,---|-/                   |   |--,
  |    |   |/                    /      |
  |    |                         \      |
  /    |                         /15k   |
  \    |                         \      |
  /R1  |                         |      |
  \    |                         |      |
  /    '----/\/\/\/--------------+------+
  |                                     |
  |                                     /
  |                                     \
  |                                     /Rs
  |                                     \
  |                                     /
  |                                     |
  '--------------+----------------------'
                 |
               ground

Conserve Ten:
I'd copy and paste that last post of yours to that other bulletin board; it sums things up in a nutshell. May I?

Sulaiman

Fri Oct 27 2006, 05:08PM

Registered Member #162 Joined: Mon Feb 13 2006, 10:25AM
Location: United Kingdom
Posts: 3141

IF you use the oil pipe as the heating element be sure to bond two sections to earth/0V and put the 'live/hot' connection inbetween or you risk galvanic corrosion of the pipe, or worse.
(I've seen this problem on trains)
a few mA leakage over time causes severe corrosion.

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