Calculation based on diameter, grids, and various arrays

[rogue0503]

Boy this is a hard one to spell out in a title. Here's what I'm trying to do.

I have a circle. Inside this circle are several rectangles that are all the same size. I also have an array that contains a matrix of these rectangles. The array has several options:

Option 1: Single row, multiple columns
Option 2: Multiple rows, multiple columns
Option 3: Single rectangle stretched across several columns (Diagonal)

Things I know:

1. Size of the Circle
2. Length and width of each rectangle
2. Array configuration (3x2, 2x8, Diagonal x7)

I need a formula that can do the following:

1. Determine how many rectangles can fit into the circle
2. Determine how many times the array will be used to touch each rectangle one time (touching outside the circle is fine, so long as the entire array is not outside the circle).

Hopefully, this makes sense. I'll try to clarify however I can.

NOTE: I have posted this question in two other forums. My apologies for cross-posting; I am new to forums and didn't know the etiquette.
http://www.ozgrid.com/forum/showthread.php?t=90395
http://www.mrexcel.com/forum/showthread.php?t=317460

[shg]

Is the array of rectangles centered within the circle?

What is your purpose in doing this?

[rogue0503]

I'm not sure if you are familiar with semiconductors, but I'll assume you are. If not, I can give you a quick crash course.

The circle is a wafer, the rectangles are the die. The array is the probe card I use that allows me to test multiple die at one time (3x2 is 6 die at each touchdown, the 7 diagonal array is 7 die tested at one time, etc.)

The objective is to see which of the various array options will result in the least amount of array touchdowns. I am looking for the most efficient pattern.

As for centering of the rectangles or the array, assume the circle is in the center of a grid. This is not always the case, but is good enough for my initial calculations. The array of rectangles typically starts at one corner of the wafer (e.g., upper left) and steps in a serpentine pattern (e.g. right to left, left to right, etc.) until it reaches the lower right corner.

[shg]

I'm not sure if you are familiar with semiconductors

In my early life, I was an FAE for Signetics before and during their acquisition by Philips ...

A full-sized probe array (touching all the die at once) would require only one touchdown, true? Or a 1/2 x 1/2 array covers everything in four touchdowns. Or a linear array that covers all the die along their short dimension across the wafer would be stepped across the other dimension ...

Doesn't there have to be some cost function that factors in?

[rogue0503]

Hello again,

I am somewhat limited to how big my array can be. I can have thousands of die on the wafer, but only have a 12 site maximum array (1x12, 2x6, 6x2, 3x4, 4x3 1x12 diagonal).

You are correct that other "cost" factors enter into this equation. This is just one of those factors. For instance, if my maximum site parallelism is 18 sites, it might actually make more sense to build a 16 site card since the 4x4 array is more square and touches down more efficiently.

I have data on actual touchdowns based on existing probe cards. I can use this data to test the accuracy of the formula. If I can get something that will work, we can use this formula when we order new probe cards for our new products and ensure we are choosing the most efficient and effective stepping pattern.

Thanks for your feedback thus far. I hope my responses are helping to paint the picture.

[shg]

I don't have any pregnant ideas at the moment, but will ponder.