KLA Tencor P-16+ - Polynomial Fit

0142530-000 AB KLA-Tencor Confidential 9-3
3/13/09
KLA-Tencor P-16+ / P-6 User’s Guide Stress (Optional feature) - Theory
Polynomial Fit 9
The Polynomial Fit uses the entire data set. It is important to note that higher order
polynomials (6th and 7th) might result in fitting data to local irregularities. The
polynomial fitting procedure is as follows:
A function y = f(x) can be expressed in terms of a polynomial order n as
y = c0 + c1x + c2x2 +…+ cnxn
As illustrated above, n + 1 coefficients exist for polynomial n. After the value of
the coefficients are computed, the new y values for different values of x can be
computed.
EXAMPLE:
In the actual polynomial fit algorithm, a 5th, 6th, or 7th order polynomial is used
for the calculation. In this example, a 3rd order polynomial is going to be used
for the purpose of illustrating the process of fitting a polynomial.
The general equation for a 3rd order polynomial is:
y = c0 + c1x + c2x2 + c3x3
To compute the coefficients 4 equations are required to compute the 4
unknowns. The 4 equations are generated by multiplying the above equation by
the coefficients of c3, c2, c1, and c0.
x3y = c0x3 + c1x4 + c2x5 + c3x6
x2y = c0x2 + c1x3 + c2x4 + c3x5
xy = c0x + c1x2 + c2x3 + c3x4
y = c0 + c1x + c2x2 + c3x3
The next step is to solve this set of simultaneous equations to find the values of
c3, c2, c1, and c0. Crout’s method is used here to solve this.
When the coefficients have been calculated, the new values for y are computed
for different values of x. The radius of curvature is calculated for any value of x
using the following formula:
where,
dY/dx = 3c
3
x
2
+ 2c
2
x + c
1
and d
2
Y/dx
2
= 6c
3
x = 2c
2
The calculated radius is then used to calculate stress using the stress formula
presented at the beginning of this section.
Rx()
1 dY()dx⁄()
2
+[]
32⁄
d
2
Y()dx
2
⁄
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=