Complex Parameterization of the Lossy Transmission Line Theory

The usual transmission line theory (TLT) found in the specialized literature often deals with its analysis for the ideal case (lossless case) or the low losses approximation, the last one obtained from a mathematical point of view. The present paper deals with a rigorous methodology that lets to address the TLT for the general lossy case in a very understanding and visual way. This methodology is based on analyzing the theory from the point of view of performing a rigorous analysis of the complex transformations (complex parameterization of the lossy transmission line theory, CTLT) which define the mathematical parameters involved in the physical description of the behavior of a transmission line with losses under time harmonic regime. From this point of view, the usual lossless case becomes a particularization of the general results in the CTLT analysis; by the way, the low losses approximation may be fully understood under this generalization as well as other interesting particular cases such as the non dispersive lossy case. One of the main important properties associated to the CTLT comes from the fact that they may be graphically represented by analytical curves which, in many cases, correspond to very interesting mathematical curves running from basic circumferences to Cassini ovals, for instance. This leads to a set of graphs that may be summarized as follows: (1) "Universal" characterizations of the behavior of both the basic parameters (characteristic impedance and propagation constant) as well as the wave parameters (impedance, admittance and reflection coefficient) involved in the transmission line theory, with all the parameterizations described by well-known analytical curves, and (2) Graphical parameterizations of losses and important physical interpretations directly induced by the geometrical properties of these curves, many of them not easily available from the usual mathematical approach. The large number of physical interpretations appearing from the graphical analysis may be exemplified by the following examples: (i) the usual Smith Chart becomes a particular case (when losses become null) of a Generalized Smith Chart. In fact, the new Generalized Smith Chart is only one of the several complex parameterizations that will be presented in the paper; (ii) the description of all the possible values of the characteristic impedance and propagation constant in terms of losses and the subsequent identification of all the transmission lines with a fixed value of a basic parameter (for instance, all the transmission lines with the phase of the characteristic impedance fixed to a specific value); (iii) from the practical and designing points of view it is possible to determine whether a solution is possible or not given some specific design constraints and, in case a solution is indeed possible, describing the range of transmission-line-wave-parameter values associated to such solution and the understanding of which parameters are susceptible of variation to resolve possible design errors, allowing optimization approaches to their solution. All these aspects are particularly relevant, both from the educational point of view, and from the standpoint of TL's based circuit design. Also, the difficulty of resolving some of the equations associated to the lossy TLT can be substantially reduced by using the geometrical properties associated to the CTLT methodology. In this sense, the paper will emphasize the concepts and physical interpretations that can be extracted from this methodology. This fully analytical methodology has also been implemented into a software tool that will be introduced also in the present paper.