4Coaxial Line: TEM Mode (cont.) zyxbaHenceNote: This does not account for conductor loss.
5Coaxial Line: TEM Mode (cont.) zyxbaAttenuation:Dielectric attenuation:Conductor attenuation:(We remove all loss from the dielectric in Z0lossless.)
6Coaxial Line: TEM Mode (cont.) Conductor attenuation:zyxba
7Coaxial Line: TEM Mode (cont.) Conductor attenuation:zyxbaHence we haveor
8Coaxial Line: TEM Mode (cont.) Let’s redo the calculation of conductor attenuation using the Wheeler incremental inductance formula.zyxbaWheeler’s formula:The formula is applied for each conductor and the conductor attenuation from each of the two conductors is then added.In this formula, dl (for a given conductor) is the distance by which the conducting boundary is receded away from the field region.
10Coaxial Line: TEM Mode (cont.) zyxbaWe can also calculate the fundamental per-unit-length parameters of the coaxial line.From previous calculations:(Formulas from Notes 1)where(Derived as a homework problem)
11Coaxial Line: Higher-Order Modes We look at the higher-order modes of a coaxial line.zyxbaThe lowest mode is the TE11 mode.xySketch of field lines for TE11 mode
12Coaxial Line: Higher-Order Modes (cont.) zyxbaWe look at the higher-order modes of a coaxial line.TEz:The solution in cylindrical coordinates is:Note: The value n must be an integer to have unique fields.
15Coaxial Line: Higher-Order Modes (cont.) zyxbaWe choose (somewhat arbitrarily) the cosine function for the angle variation.Wave traveling in +z direction:The cosine choice corresponds to having the transverse electric field E being an even function of, which is the field that would be excited by a probe located at = 0.
16Coaxial Line: Higher-Order Modes (cont.) zyxbaBoundary Conditions:Note: The prime denotes derivative with respect to the argument.Hence
17Coaxial Line: Higher-Order Modes (cont.) zyxbaIn order for this homogenous system of equations for the unknowns A and B to have a non-trivial solution, we require the determinant to be zero.Hence
18Coaxial Line: Higher-Order Modes (cont.) zyxbaDenoteThe we haveFor a given choice of n and a given value of b/a, we can solve the above equation for x to find the zeros.
19Coaxial Line: Higher-Order Modes (cont.) A graph of the determinant reveals the zeros of the determinant.Note: These values are not the same as those of the circular waveguide, although the same notation for the zeros is being used.xn3xxn1xn2
20Coaxial Line: Higher-Order Modes (cont.) Approximate solution:n = 1Exact solutionFig from the Pozar book.
21Coaxial Line: Lossless Case Wavenumber:TE11 mode:
22Coaxial Line: Lossless Case (cont.) At the cutoff frequency, the wavelength (in the dielectric) isCompare with the cutoff frequency condition of the TE10 mode of RWG:bsoaor