## Curve Fitting finds defective parts.

submitted by: Phil Brubaker
on Wed Nov 23 14:10:33 2005

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Course: Calculus
Topic: Lorentzian Model of Isolated Pulse from Disc Drive
Resource type: Tool
Catalogue entry:
http://mathforum.org/mathtools/
Resource location:
http://www.digitalcalculus.com/demo/curvfit.html
Story:

This is a story of how a math model located a manufacturing flaw.

## Modeling Digitized Signal from Magnetic Recording

Magnetic recording of transitions written onto a computer disc drive may produce an isolated pulse as shown below. This pulse comes from a disc drive's read/write channel.

Figure. An isolated Readback Pulse from a disc drive

The signal's shape is very important to the electrical engineering development groups of disc drives. A readback pulse should be symmetric and have a relatively fast rise time (i.e. sharp slope) for improved peak detection capability. A math model for the pulse can help gain insight into what electronic sub-system/components are causing the pulse to be asymmetric or have a slow rise time.

The longitudinal magnetic force was assumed the main contributing factor in determining a readback pulse shape, before the early 1980's. This force component was modeled by a series of three Lorentz functions. These functions have varying independent parameters that are dependent upon the drive's Thin-Film-Head (TFH) composition, size and shape. ***Note: The value for these parameters was helpful in understanding a design and manufacturing flaw.

A Lorentz function has represented/modeled an isolated readback pulse for some time. The basic Lorentz function is defined as y = 1 / (1 + y^2). The isolated pulse model is a composite of three Lorentz functions as stated here:
y = v_i / [ 1 + x_i^2) where x_i = 2(t - t_i)/ pw_50_i

where v_i = Amplitude of a Lorentz pulse;
pw_50_i = Lorentzian pulse width, measured at 50% height of v_i; and,
t_i = Origin of the ith Lorentzian.
i = 1 to 3

Real-world data fit well and when a TFH produced a bell curve with a step on its back side, it too fit well. That told us that this model was excellent and the step was real. Such a TFH had some alignment problem in manufacturing and thus was destroyed. Until this model was tested these bad TFHs were had to find and demolish.

A good math model is worth its weight in gold!

Send comments to Phil Brubaker at phil@digitalCalculus.com