Non-linear regression of Curie's Law

[jdstokes]

Hi,

I've collected some data on the relative permeability of ferrite at various temperatures, subject to a constant external magnetic field and I'd like to fit a curve to the data.

I believe that stat-mech theory predicts that $\mu_r = 1 + a\tanh(b/T)$ where $T$ is thermodynamic temperature. The constant $a$ is clearly $\max \mu_r -1$, but I can't figure out what condition I should use to estimate $b$?

Any help would be greatly appreciated.

James

 

[LeonhardEuler]

I'm no statician, but I've come across this type of problem before. I'll tell you how I dealt with it and you can decide if its reasonable (which I think it is).

You want to choose b in a way that minimizes the "error". If you have N measurements of mu as $\mu_1, \mu_2, ... \mu_n$at temperatures $T_1, T_2, ... T_N$ then you can define an error by:
$$E=\sum_{i=1}^{N}(\mu_i-\mu(T_i,b))^2$$
Where:
$$\mu(T,b)=1 + a\tanh(b/T)$$
You want to find the value of b that minimizes the error so you come out with:
$$\frac{\partial E}{\partial b}=0=2\sum_{i=1}^{N}(\mu_i-(1 + a\tanh(b/T_i)))(\frac{a}{T_i}sech^2{\frac{b}{T_i}})$$
Now this equation should not be too difficult to solve numerically as long as the number of measurements is not gigantic. You could use goal seek in excel. One thing to whatch out for, though: hopefully the solution is unique. If it is not you have to find the minimum amoung all the solutions. Excel will not tell you if there are other solutions. Maybe you can show mathematically the solution is unique, I haven't thought too much about that.

Also, be warned: I didn't read about this in a book, I just had a similar problem and this method made sense to me. Use it at your own discretion.

 

[jdstokes]

I think this is a good idea, numerical solutions to such things can be found easily using Mathematica.

Thanks

James