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Engineering Mathematics - Investigation of practical applications of complex mathematical concepts using MATLAB Assessment

Purpose of the assessment - The purpose of this assignment is to investigate practical applications of complex mathematical concepts and gain understanding of how mathematics is used in the engineering context. At the completion of this unit students are expected to be able to:

a. Develop problem-solving skills in the context of engineering mathematics;

b. Have a practical understanding of mathematics in the engineering context;

c. Gain experience using an extended range of practical applications of complex mathematical ideas;

d. Apply knowledge of basic science and engineering fundamentals to real-life situations;

e. Communicate effectively, not only with engineers but also with the community at large;

f. Undertake problem identification, formulation and solutions;

g. Solve a broad range of problems in mathematical areas;

Assignment Description -

This is a group project, requiring students to work in groups of 3 students. Team members are expected to equally participate, and collaboratively work towards the completion of the project.

This task requires you to research and try to understand the given topic. Modern mathematics requires the use of computational tools in order to solve difficult real-world problems. Tools are required because often the problems are either too large or not possible to solve analytically, and building physical models is impractical. Using computational tools is additionally often an open-ended exercise, where some formal theory is implemented, possibly with the aid of pre-existing analysis or models. This assignment gives you a chance to investigate into application of complex mathematical concepts and gain hands-on experience in how MATLAB is used as a problem-solving tool. The Engineering Mathematics unit assists you to become familiar with the mathematical skills required to solve engineering-related problems. The mathematical skills acquired in this Engineering Mathematics unit will provide a necessary background to understand the algorithms and methods used in this document.

You need to

• Show how well you have understood the problem and simulations or calculations in this system and explain how MATLAB is used as a problem-solving tool.
• The projects are open ended in nature. As long as your program can perform the assigned tasks, there will be no correct or wrong approaches. Certainly, there will be more acceptable and attractive solutions in comparison with competing solutions.
• In the event you cannot complete the task, you should turn in whatever you have completed and grade will be based on completed work. Bottom line will be, if we cannot get your program to execute, it will be graded based on what you have completed. In all cases it will be essential that you submit a complete set of files to test your program. It will also be important to give clear instructions of how to run your program. This could be done in various ways. One good way to document how your program executes is to prepare "readme.txt" file.
• Each group will interactively present its solution in Week 12 by demonstrating how the code is executed.
• All members in a group must participate in the presentation and must have a reasonable familiarity with the project, even if they have not been the lead person on that specific topic.

Antenna Radiation Pattern Calculation -

Antenna radiation pattern is a graphical representation of the power radiated by antenna relative to direction. Antennae radiate differently in different directions. Usually the power is measured far away from the antenna at a fixed distance.

Part A: Dipole antenna radiation pattern

Write a program using MATLAB software to plot the radiation pattern of a dipole antenna.

Use radiation frequency of 950 MHz.

• Calculate wavelength of radiation λ = c/f, where c=3*10⁸ m/s is the speed of light.
• Calculate k=2*π/λ.
• Assign observation point to be R=10* λ from the antenna.
• Assign I₀ = 5 Amperes.
• Use length l of antenna that will be assigned to each group in the classroom.

Create an array of angles θ from 0 to 359 degrees or 0 to 2π radians.

Use formula (2) to calculate radiated power S for each angle direction θ.

Normalize the power by dividing all values of S by maximum power S max as shown in formula (3).

Plot normalized power S_n versus angle θ using "polar" plot function. Add title to the plot. Save the plot.

Part B: Dipole antenna electric field

Write a program to plot the electric field lines of a dipole antenna from Part A using MATLAB's "contour" plot. Use the built-in Matlab help function to find an example of how to use the "contour" function. Look up "linspace" and "meshgrid" commands.

• Assume the antenna is located at the center of the plot. Assume x and y values are in rectangular coordinates from - 2*λ to +2*λ.
• Calculate electric field E_θ using formula (1), but calculate R=√(X²+Y²) , sinθ = Y/√(X²+Y²) and cosθ = X/√(X²+Y²) first, this convert polar coordinates to rectangular coordinates and with get you values of E on all points of X-Y grid.
• Take real part Re(E_θ) of the complex result for the field.
• Use "contour" function to plot electric field lines for the radiating antenna. Add title to the plot. Save the plot. Example electric field lines plot for l= λ/2 is shown in Figure 4 (colours may be different in your plot).

Part C: Dipole antenna total radiated power calculation

Write a program using MATLAB software to calculate the total power radiated by dipole antenna using integration:

• Use the same radiation frequency, wavelength of radiation λ, k, l, I₀ and observation point R as in Part A.
• Modify the code from Part A to represent power S as a function of angle θ.
• Calculate integral for total radiated power Prad given in formula (4).

Submission guidelines:

1. Each team will submit one project report that must be a PDF or word file. Your MATLAB codes can be submitted as a text attachment in your project report or as a separate .m files and should contain everything necessary to run the file.

2. Prepare a report summarizing your findings. Your report should include the following:

• The cover page must identify students (name and number), teaching staff, and assignment.
• Body of the report should include

i. Introduction -In this section, you need to give a brief overview and your understanding of the problem discussed in the document and outline of the report.

ii. Methods -You need to describe your understanding of methods you used for simulations or calculations in MATLAB. Briefly explain the key files and functions in your code.

iii. Results and Discussion- Present quantitative and qualitative results of your approach and discuss the findings.

iv. Conclusion - Summarise the findings.

v. References - Sources must be cited in the text of the report, and listed appropriately at the end in a reference list.

3. Prepare a PowerPoint presentation for 5-10 minute presentation. No more than 7-8 slides. You must present as a group, every member of your group needs to speak. You must have visual aids to support your presentation and you must acknowledge the source of the information you present. The presentation should focus on the Mathematics/MATLAB.

4. The assignment must be submitted in soft (electronic) copy under Moodle. The MATLAB program files and presentation powerpoint file should also be uploaded. The pages of the assignment must be clear on each page.

Section to be included in the report -

Coding - Executable MATLAB code, A complete set of files to test your program.

Presentation - Marks for the Presentation will be based on the material provided, presentation skills, and understanding of the material.

Report -

Introduction - Brief overview and your understanding of the problem.

Methods - Describe your understanding of simulations or calculations. Briefly explain the key files and functions in your code.

Results - Present your results.

Discussion - Discuss the results.

Conclusion - Concluding remarks.

Note - All figures and formulas are in attached file.

Attachment:- Assignment File.rar