I have an issue, of which I'm not sure it can be solved, with or without Mathematica. I feel as if it should be possible, but I'm quite clueless as to how.
Some introduction is required. I'm working in a research lab, and we are trying to characterize some samples that were fabricated. These are chips with capacitances on them, three different types. Lets call them $C_1$, $C_2$ and $C_J$. What we then have is 5 devices on which $C_1$ and $C_2$ are constant and $C_J$ is varied, and 4 devices on which $C_J$ is constant and $C_1$ and $C_2$ are varied. However, this is fabrication, we are not sure of the exact values of these parameters. This is what the experiment wants to characterize. They are both being varied by a parameter which we do know: the number of fingers we write on the chip, lets call them $F_{1,2}$ and $F_J$. So the idea is to find the functions $C_{1,2}(F_{1,2})$ and $C_J(F_J)$. Moreover, and this is a little messy, in these measurements it is the case that $C_1$ and $C_2$ always consist of the same amount of fingers, they simply have a slightly different length and thus lead to different capacitance values.
On these devices, measurements are performed. However, we can't measure the capacitance directly; we measure some energy related parameters, $\omega_{1,2}$ and $J$. This is where it gets a bit tricky, because they have a complicated definition, which might be easiest to just give in code:
Cmatrix = ( {
{C1 + CJ + CJ, -CJ, 0, -CJ},
{-CJ, C2 + CJ + CJ, -CJ, 0},
{0, -CJ, C1 + CJ + CJ, -CJ},
{-CJ, 0, -CJ, C2 + CJ + CJ}
} );
InvCmat = Inverse[Cmatrix];
ω1 = Sqrt[1/L InvCmat[[1, 1]]];
ω2 = Sqrt[1/L InvCmat[[2, 2]]];
J = 1/2 InvCmat[[1, 2]]/Sqrt[InvCmat[[1, 1]] InvCmat[[2, 2]]] Sqrt[ω1*ω2];
I should add that in the above L is a known parameter, with value 1.84.
So, what I have is essentially nine datapoints:
{{20, 20, 2, 7.70438, 7.50004, 0.156103}, {20, 20, 3, 7.60618,
7.30572, 0.212588}, {20, 20, 4, 7.46063, 7.18521, 0.273479}, {20,
20, 6, 7.25941, 6.79769, 0.354176}, {20, 20, 10, 6.93511, 6.59471,
0.494317}, {13, 13, 3, 8.8022, 8.20026, 0.297579}, {16, 16, 3,
8.22736, 7.78623, 0.252157}, {20, 20, 3, 7.58361, 7.359,
0.212081}, {24, 24, 3, 7.08458, 6.90351, 0.178109}}
Each each containing (in order of occurance) $F_1$, $F_2$, $F_J$, $\omega_1$, $\omega_2$, $J$. From these datapoints I'd like to find a model for $C_i(F_i)$ and $C_J(F_J)$. Maybe an exact model is not possible, but an approximation should be. Based on physical intuition the relationships should be linear, but it might get a little messy due to fabricational issues.
So, given all this, would anyone have an idea on how to start?
I should also add that the fact that $C_1 \neq C_2$ is an error in the fabrication. Ideally this would not be the case, which simplifies the problem for sure, but even then I'd not know how to do such a fitting routine. So if instead you want to set $C_1 = C_2$ in the initial matrix and describe a method from there it would also be valuable.
Edit: I forgot to add. All values of $C$ are positive and real and $probably$ on the order of $10^{-4}$.
J
onlySqrt[InvCmat[[1, 1]]
is in the denominator. All the rest of the terms are in the numerator. Can you validate that this was your intention. If you want all of the terms to the right of the/
sign to be in the denominator you need to put parenthesis around them. $\endgroup$ – Jack LaVigne Sep 1 '15 at 19:48