I have generally picked subjects for articles as the topic of the moment, depending on my current program development or consulting tasks. I will be spending the next two months thinking almost exclusively about Aether, and continuing announcements about the program would become redundant. As an alternative, I decided to write a continuing series of articles discussing worked solutions for SATE (Static-field Analysis Toolkit, Educational), our freeware package for electromagnetic education. The exercises follow the lists in the posts http://www.fieldp.com/myblog/?p=33 and http://www.fieldp.com/myblog/?p=34.
As a preliminary, I wanted to say a few words about opportunities for using SATE. We would like to work with you incorporating the package into classes, computer laboratories and publications. Please contact us at techinfo@fieldp.com to arrange a site license for your school. SATE is also available for inclusions if you are publishing a book with an accompanying CD. There is no charge for either application, but we do require an agreement for citations and distribution. You can download SATE at no charge at:
http://www.fieldp.com/sate.html
In this example, we shall use EStat to find the capacitance/length of the two-wire transmission line shown in the figure below. The wires have diameter D and the center-to-center distance S. The medium has relative dielectric constant ?r. The theoretical value is:
c = pi*?r*?0/invcosh(S/D).
Here are links to download the EStat input files that I prepared:
http://www.fieldp.com/myblog/examples/twowire_estat.min
http://www.fieldp.com/myblog/examples/twowire_estat.ein
The parameters for the example are D = 0.50 cm, S = 2.0 cm and ?r = 1.871. The wire voltages are -0.5 V and +0.5 V. The problem is that the theoretical system extends to infinity while numerical calculations are limited to finite solution volumes. Fortunately, |E| of the dipole array drops rapidly with distance from the wires, so we can get a gopod approximation by surrounding the wire region with a large volume with coarse element resolution. The outer boundary is set to ground potential (0.0 V). The figure shows a detail of equipotential contours near the wires. The inset shows the full solution volume with the view area outlined in brown.
The calculated field energy/length is u = 1.271E-11 J/m. With the applied potential difference of 1.0 V, the capacitance per unit length is c = 2u = 2.542E-11 F/m. The value predicted by the above formula is 2.522E-11. The small relative difference of 0.8% results mainly from the finite size of the solution volume.
Here are some additional calculations you can try:
- Apply symmetry boundaries and limit the calculation to the space x > 0.0, y > 0.0. Remember to multiple the field energy by 4.0 to determine the capacitance.
- Find the capacitance/length of a shielded two-wire transmission line. Here, the wires are enclosed in a grounded cylinder of radius R. Assume that the wires have voltages +V/2 and -V/2. Compare the capacitance to that for the unshielded wires for different values of R. (Note: because there is no simple analytic formula for capacitance/length in this geometry, a numerical calculation is the best approach.)
- Find the capacitance per unit length of a coaxial transmission line and compare your results with theory. Find the change in capacitance when the center conductor is offset (somebody rolled a scope cart over the cable).
