University of Connecticut Fall 2019
ECE 3431/CSE p3802 Numerical Methods in Scientific Computation
Dept. of Electrical and Computer Engineering DT
Programming Project #1 Due September 23, 2019
Background : In the wire and cable industry, coaxial cable is a very important product. A basic coaxial cable is depicted below1
We desire to know when the cable will carry a single RF mode (a mode is a unique solution of the Maxwell equations that obeys the boundary conditions.) Each mode has a different phase velocity and so having multiple modes propagating on the cable would lead to unwanted effects in practice.
However, when we solve the Maxwell equations for this geometry, we find that one set of the cutoff wave numbers satisfy a transcendental equation:
Here:
• J0 is the Bessel Function of the First Kind of order zero
• Y0 is the Bessel Function of the Second Kind of order zero
• a is the outer radius of the center core in inches
• b is the inner radius of the metallic shield in inches
• k is the cutoff wavenumber in radians/inch = the thing we seek!
The lowest value of k determines the cutoff frequency for the first higher order mode; the second value of k determines the cutoff for the second, etc.
1 Wikipedia public domain image ECE 3431/CSE 3802 Project #1
(1)
Project Objective : Your mission in this project is to implement the different methods we’ve discussed in class for solving equations of one variable to this problem. For the cable in this project, please use b = 0.325 inches and a = 0.075 inches. You should use your coded routines to find the first four values of k>0 that satisfy equation (1).
To code the Bessel functions, please feel free to use the following partial sums; we will derive these later on in this course.
Here ge is the Euler constant (approx. value 0.5772156649). Sufficient accuracy for this project can be obtained from these partial sums by setting N=10.
Please attempt to solve the cutoff equation using at least four different numerical methods. Be sure to verify that your code works by testing it on a simple trial case. You can test your routines for the Bessel functions using data supplied by DT on the course website.
Expected Output :
Please document your results in some sort of graphical summary form – NOT a formal written report. You can make a Powerpoint presentation, or a poster summarizing the work, or some other visual format. Be creative. Imagine that you’re going to be briefing your project to a management team that is interested in the results – so be sure to explain (at a minimum):
• How you tested the code to make sure it works
• How fast each of the methods was able to find the solution
o Can you estimate the rate of convergence?
• Which method(s) appear the best suited to this problem
• Are any methods unstable or potentially unstable and does this affect the results?
• Which method(s) if any are not well suited to this problem?
ECE 3431/CSE 3802 Project #1