Mcmillan Mcgee

Problem Statement

My mathematical problems are centered on solving certain boundary value problems involving Maxwell’s equations and linking these to the laws of thermodynamics. As you may recall, my team and I at McMillan-McGee have developed induction heating technology for in situ soil remediation. As such, we have developed heater casings, work coils, and inverters which generates high frequency alternating current. We have also developed a great amount of the analytical work for engineering this equipment.

An important aspect of our inverter design (or for that matter any high frequency inverter in general) is to have a good understanding of the electrical properties (resistance and inductive reactance) of the DC bus bar that supplies current to the high speed switching devices (IGBTs¹ and SiC Mosfets²). Knowing this would allow us to design a suitable bus bar system³ that can absorb energy caused by switching transients from these semiconductor devices as a result of commuting current through the work coil. The bus bar is an electrically conducting (copper) rod of a certain length with rectangular cross-section. We would like to be able to compute in closed form⁴ the bus high frequency effective resistance⁵ and inductance. An approach that I am presently using is a technique that I have learned from Norman McLauchlan^6 where he approximates the rectangular cross-section of a bus bar with an ellipse having suitable dimensions. In McLauchlan’s textbook he only computes the effective resistance of a long straight conductor of elliptical cross-section to high frequency alternating current. This spring, I extended his work to compute the inductance too. The big problem with McLauchlan’s work is that he is working in the Gaussian system of electromagnetic units; not, the MKS as most electrical engineers are familiar with including myself. I immediately became aware of this when I found that McLauchlan uses unity for the permeability of copper (same as that for free-space). So, I place some doubt on the my interpretation of his result as well as my extension for computing inductance. It should be noted that when passing sufficiently high frequency alternating current on a bus bar of rectangular (approximated to elliptical) cross section the current is confined mainly to a surface layer of the conductor, and the current density is negligible within the body of the bus bar. Thus the component of the magnetic field normal to the bar’s surface tends to rapidly diminish immediately above it. Therefore, the magnetic field can be assumed tangential to the bus bar. It then follows that on the surface the vector magnetic potential, A, is constant and satisfies the same conditions as the scalar electrostatic potential, “phi”. Consequently, McLauchlan draws the conclusion that the surface distribution of current density is identical to that of a bar holding an electric charge. Hence, the total current flowing axially on the bus bar corresponds to the total surface charge. With these assumptions, McLauchlan developed formulas involving a certain elliptic integral for computing current density, power loss, and the high frequency resistance.

To tackle this problem requires a team familiar with electrostatics and electrodynamics. Certain individuals within the team should possess a working knowledge on curvilinear coordinates, with special attention to the system of orthogonal elliptic-hyperbolic coordinates. Also, team members should be familiar with elliptic integrals. Fortunately, McLauchlan’s solution to the problem side-steps the task of solve a boundary value problem using Mathieu functions.

References