GoldExI : FAQ : How Ic, Rn, Ie algorithms work?

Question

How I_c, R_n, I_e algorithms work?

Answer

General notes.

I_c, R_n, I_e denote "critical current", "normal resistance" and "excess current" correspondingly.
All algorithms take into account whether IVC is one-, two- or more-directional. For many-directional curves it is the 2nd branch which is usually processed. If the curve is too complex or too short, the determination of number of branches (turning points) and "good" axis is impossible. As a result the whole algorithm will not work.
All algorithms work well regardless of I-axis orientation i.e. I-axes can be x as well as y. I-axis is defined as one containing "A" (or its multiples such as mA, uA) as units.

Calculating I_c.

GoldExI scans the curve from the first point and find the point with V<0
it scans further and find a point inside criterion
it scans further and find a point outside criterion
the current in the previous point is I_c.

Calculating R_n and I_e.

The simple (old) method.

connect the first and the last points of one-directional IVC, or the first point and the first turning point of multi-directional IVC
the slope of this curve is R_n, the excess current I_e is always zero.

Whole Curve Least Square Fit (LSF) method

GoldExI calculates LSF using all points of the curve. Optionally you can assign a weighting coefficient proportional to voltage.

Converging Least Square Fit (LSF) method

Typical value of current and resistance are measured. As such GoldExI takes the values at the last point of one-directional IVC or of 1st point of the 2nd branch of multi-directional IVC.
GoldExI takes several "starting points" and pass a straight line through them using LSF. For one-directional curve this points are the last points of the curve, for multi-directional IVC, those points are the first few points of the 2nd branch of the curve.
The slope of this straight line is R_n, and the shift of this line is I_e.
Then GoldExI adds the next point of the IVC to the above group of starting points and makes LSF again.
The above procedure is repeated by adding more and more points to the group until the series of so-calculated R_n AND I_e values converges. The convergence criterion is:
| X_i+1 - X_i | / X₀ < RelativeAccuracy
where X is either R_n or I_e, and X₀ is the typical value of R_n or I_e which we have found in the beginning.

The Converging LSF method has two parameters "starting number of points" and "relative accuracy" which can be specified before applying the above procedure.

Where these algorithms are used?

The above algorithms are used separately or together in:

Math | Histograms | Histogram Ic
Math | Histograms | Histogram Rn
Math | Histograms | Histogram Ie
Math | Ic, Rn, Ie table

Before applying each algorithm the user is asked about necessary parameters.

In the case of Math | Ic, Rn, Ie table, the user can also request to draw Rn-lines to visually control calculation. The Rn-line is a straight line which consists of 3 points. For the case of converging LSF method these points are:

the point from where LSF has started
the point where convergence was achieved
the point on Rn-line with x-coordinate equal to the x-coordinate of the first point of the curve.

For the simple method these points are:

the first point of the curve
the middle point of the curve
the last point of the curve

For the Whole Curve LSF these points are:

the left-most point
the middle point
the right-most point

The end.