This assignment forms a marked portion of the mathCAD section of your PH15010 module.

Create a mathCAD worksheet to hold the answers to all of the questions.

All formulae and numerical constants required to answer the questions are either contained within this sheet or may be found in the mathCAD resource center.

Your completed worksheet should be emailed as an attachment to dpl@aber.ac.uk before 23:59 on Saturday 13th December 2003. If you have difficulty sending files as attachments, please contact me before that date.

Marks will be given for correctness of answers, proper use of mathCAD facilities, appropriate comments and explanations, layout of the sheet and style.

## Problem 1 – Moore’s Law

In 1965 Gordon Moore, the founder of Intel, proposed that the complexity and component count of silicon integrated circuits would double every 18 months. This prediction has turned out to be startlingly accurate and has driven the electronic revolution we see around us.

The following data file is taken from Intel’s web site and contains two columns of data. The first column has the year of introduction of various processors and the second has the number of transistors in each of these processors.

http://users.aber.ac.uk/dpl/ph15010/handouts/Transistors.txt

Copy the file into your own filestore.

Create a file read component to read in the data file and split it into two appropriately named vectors.

Create a plot showing the number of transistors on a chip over the years 1970 to 2000. Use a logarithmic scale to show how this approximates to a straight line.

By taking logarithms and performing regression analysis, show how close to the target doubling time of 18 months Intel’s engineers were able to achieve over the 3 decades from 1970 to 2000.

Create a function Moore(y) which will predict the number of transistors on a chip for any given year.

Create a plot showing how this function compares with the experimental data.

## Problem 2 – Detector Data

This exercise is taken from Example sheet 8. If you have already completed the exercise, you can simply copy and paste the exercise into the sheet you are submitting for this assessment.

I have prepared two files for you to bring into MathCAD and perform some simple analysis upon. The files are data taken from one of my detector experiments. The first column shows the detector number, and the second column the number of counts received on each detector over a 10 second period. The two files are nominally from the same spectrum but shifted along the array.

On a blank MathCAD worksheet, select “Insert|Component…” from the menu. This will bring up a wizard where you should first indicate that you want to read data from a file. On the next page of the dialog box you should leave the file type as ‘Text File’ and use the browse button to locate the datafile you have stored on your M: drive in the previous step. Once the wizard closes, your worksheet should show an icon representing a file read component, with a little disk on it. The cursor will be positioned in a placeholder ready to hold the name you wish to give the dataset. Once you have given it a name your worksheet should look like this:

Now, using the techniques we started to explore with the resistor experiment and the photoelectric effect, carry out the following analyses on the data. I have put some hints at the bottom of the page which may help if you get stuck.

• From the matrix of data, create a vector to hold the Detector number and another to hold the counts.
• Plot the counts against detector number.
• Create a plot of count rate vs detector, given that the sample period was 10 seconds.
• Each detector is 20um x 2mm. Produce another graph with the plot rate scaled in terms of counts per second per cm2.
• There are 2 clear peaks in the data. Find the height of the largest peak in terms of counts per second.
• Create an expression which will give the number of the detector with the highest counts.
• Use the submatrix function to create a matrix containing the data from the highest counting channel together with the 6 channels on either side.
• Calculate the total number of counts in the highest peak and its adjacent channels that you have just created above.
• Create a function called PeakArea which takes as its arguments the channel number and the width of the peak in channels. It is to return the total number of counts of the given channel together with those up to 1 width away. For example, PeakArea(42,3) will give the total counts from detectors 39,40,41,42,43,44 and 45.
• Use the PeakArea function to re-calculate the area of the highest peak and the 6 channels either side.
• Create a graph which will show the largest peak and a specified number of adjacent channels on either side.
• Go back to the file read component and right click on it, select “Choose File…” from the pop-up menu and pick the file “DetectorData2.txt”. Click out of the file read component and check that all of your answers and graphs change to reflect the new data file.

### Hints

The following hints may be useful to you. Use the help system and conduct some experiments with MathCAD to see how.

You may want to create a unit for the micron (1 um is 10-6 m)

The max() function will tell you the largest number in a vector

The csort() function can be used to sort a matrix so that a particular column is in ascending order.

You can reverse() the results of a sort.

You can use variables in the placeholders which determine the axis scaling of a graph.

The submatrix() function can be used to pick off part of a matrix.

The matrix toolbar has one way of summing the elements of a vector.

The calculus toolbar has two more ways of doing summations.

## Problem 3 – Symbolic Maths

Consider a rectangle with sides of length a, and b.

Using the symbolic processor in MathCAD show that the area of the rectangle is maximised when a=b, for a given perimeter.