Abstract
Some modal (or decoupled) transmission-line properties such as per-unit-length impedance, admittance, or characteristic impedance have long been held to be, in general, non-unique. This ambiguity arises from the nature of the similarity transformations used to relate the terminal and modal domains, for which the voltage transformation matrix has been shown to be only loosely related to the corresponding current transformation matrix. Modern methods have attempted to relate the two, but these relations typically rely on arbitrary normalizations, leading to strictly incorrect and/or non-unique results. This work introduces relations between the two transformations, derived from the physical equivalence of total power and currents between the two domains, by which the transformation matrices can be unambiguously related to each other, and the modal properties uniquely solved. This technique allows for the correct extraction of the modal transmission-line properties for any arbitrary system of conductors. Multiple examples are studied to validate the proposed solution process.
Highlights
T RANSMISSION-line (TL) theory is a powerful concept that allows the fields of transverse electromagnetic (TEM) modes to be expressed as unique voltages and currents, making it a critical tool for high-accuracy circuit design
It was claimed that modal values could not be determined from terminal-domain equivalents, fundamentally due to the complex nature of the transformation matrices [11]. While this claim is surely valid if the transformation matrices are assumed generally complex, it is further demonstrated in this work that TL modes – under the assumption of transverse electromagnetic (TEM) propagation – are strictly normal, which allows these matrices to always be expressed as entirely real [3], [12]
This assumption gives rise to the TEM approximation, in which results derived for strictly ideal TEM cases may be applied to realistic, quasi-TEM TL modes
Summary
T RANSMISSION-line (TL) theory is a powerful concept that allows the fields of transverse electromagnetic (TEM) modes to be expressed as unique voltages and currents, making it a critical tool for high-accuracy circuit design. The main contribution of this work is to demonstrate that any such ambiguities can be resolved by noting that physical quantities – total current – must be respectively equal in both domains, since the transformation between them is a change of basis These physical constraints are used together to eliminate the unknown eigenvector scaling factors to within a sign and produce modal properties. It was claimed that modal values could not be determined from terminal-domain equivalents, fundamentally due to the complex nature of the transformation matrices [11] While this claim is surely valid if the transformation matrices are assumed generally complex, it is further demonstrated in this work that TL modes – under the assumption of transverse electromagnetic (TEM) propagation (referred to hereon as the TEM approximation) – are strictly normal, which allows these matrices to always be expressed as entirely real [3], [12]. The properties at present may only be determined through laborious analytical solutions or extraction from the scattering of a finite array of structures [18], [19], where it may be excessively difficult to properly excite each mode and a small error will always be present due to the finite extent of the array
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