## Intuition For the Steinhart-Hart and B/Beta-Parameter Equation

hiimbob
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### Intuition For the Steinhart-Hart and B/Beta-Parameter Equation

So I've been doing detector building for a bit, and I've gotten used to using the Steinhart-Hart equation. I understand that it makes predictions that match with experimental data, but I don't really have an idea of where the equation comes from.

To me, approximating the reciprocal of temperature as a depressed cubic in terms of the natural logarithm of resistance feels like a random approximation to choose (why can't it be the reciprocal of temperature squared? why can't it be a quintic polynomial in terms of the reciprocal of resistance?) and I was wondering if there's any physical principle that the formula is derived from.

After a bit of looking around on the internet, the closest thing I got to a derivation was on the Wikipedia article for the Steinhart-Hart equation which says "The most general form of the equation can be derived from extending the B parameter equation to an infinite series". Unfortunately, the B-parameter equation seems to be equally unmotivated, as I can't really find any derivations of it.

Here's the explaination given on the wikipedia article and the questions I have about it:
The B-parameter equation is $R=R_{0}e^{B\left({\frac {1}{T}}-{\frac {1}{T_{0}}}\right)}$. Re-arranging the terms of the B Parameter equation gives ${\displaystyle {\frac {1}{T}}={\frac {1}{T_{0}}}+{\frac {1}{B}}\left(\ln {\frac {R}{R_{0}}}\right)=a_{0}+a_{1}\ln {\frac {R}{R_{0}}}}$. Following this, the Wikipedia article says that this equation can be expanded further by using the infinite series ${\displaystyle {\frac {1}{T}}=\sum _{n=0}^{\infty }a_{n}\left(\ln {\frac {R}{R_{0}}}\right)^{n}}$. From this, taking $a_2=0$, $R_0=1$, and only using the first 4 terms (ends up being 3 terms since one is 0) of the series gives the Stienhart-Hart equation.

First of all, I'm unaware of any derivations of the B-Parameter equation. The form of the equation looks like it comes from an IVT problem, namely the equation $\frac{dR}{dT}=-\frac{BR}{T^{2}}$ with initial condition $(T_0, R_0)$; however, I'm not sure where this specific differential equation would come from. Secondly, I don't understand how the equation for the reciprocal of temperature can be expanded to an infinite series, the series and original expression seem like 2 completely different things (yes the first couple of terms of the series match with the original, but where to the infinite number of other terms come from?). And lastly why is $a_2=0$? I understand that the reference resistance can be any arbitrary value because the constants can be shifted (by using the properties of logarithms and expanding each of the powers of the binomials) but I don't see any similar reasoning to setting $a_2=0$.

Any insight about these questions or about the derivations of the equations would be greatly appreciated.
Thanks!
Last edited by hiimbob on February 23rd, 2021, 5:40 am, edited 1 time in total.
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### Re: Intuition For the Steinhart-Hart and B/Beta-Parameter Equation

This is really a quite awful explanation, but some good starting points are first identifying 1/T and the presence of ln(x).

Since thermistors are semiconductor devices, we can compare them to the Shockley Diode Equation: https://en.wikipedia.org/wiki/Shockley_diode_equation.
Notably, there's an exponential to the power of (some factor) over temperature. Solving for temperature gives ln(current) ~~ 1/T, which matches closely to the behavior of a thermistor.
After setting the parameters of the equation to 1/T ~~ ln(R), the Steinhart-Hart becomes a simple polynomial approximation -- a fairly reasonable standard to use.

The a_2=0 thing is just an approximation, real-life thermistors often have a very low value for it, so many examples ignore it. When I built detector, I included it in my model; it doesn't make that much of a difference.

The use of a polynomial approximation essentially helps model many of the "oddities" of an individual thermistor, since it's not an ideal element. Other approximations would probably work fine as well after using 1/T and ln(x) to account for most of the characteristics of the device.
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