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Understanding Spherical Coaxial Transmission Lines: Boundary Conditions and Properties

This note discusses spherical coaxial (TMr) transmission lines, focusing on boundary conditions, mathematical modeling, and properties. It highlights how the boundary conditions are set to zero, leading to a transcendental equation for eigenvalues that must be solved numerically. The note compares potential distributions satisfying the boundary conditions on the cones, including the radial and angular electric fields. It also briefly touches upon the differences between spherical and cylindrical coordinates within the context of transmission line theory.

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Understanding Spherical Coaxial Transmission Lines: Boundary Conditions and Properties

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  1. ECE 6341 Spring 2014 Prof. David R. Jackson ECE Dept. Notes 23

  2. Spherical Coax z y x Note: If 2 >  / 2, we have a bicone. TMr

  3. Spherical Coax (cont.) Boundary conditions set to zero

  4. Spherical Coax (cont.) Hence Normalizing by multiplying by , we have

  5. Spherical Coax (cont.) (The superscript “D” stands for “double cone.”) This is a transcendental equation for : (These values must be found numerically.)

  6. Spherical Coax (cont.) z y x TEMrtransmission-line type of mode Start by assuming Ar with m= 0: From handout:

  7. Spherical Coax (cont.) TEM property Choose (TEMr) Note: This potential satisfies the boundary conditions on both cones, since Er= E = 0.

  8. Spherical Coax (cont.) Examine the fields that come from the term Note that Hence, we don’t need to consider P0.

  9. Spherical Coax (cont.) Choose

  10. Spherical Coax (cont.) Hence

  11. Spherical Coax (cont.) Note: The constant in front will be ignored.

  12. Spherical Coax (cont.) z y x Note: To compare with cylindrical coordinates, we can use

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