FDTD formulations and node-to-node admittance functions models for time domain analysis of frequency dependent lossy multi-conductor transmission lines

The multi-conductor transmission line (MTL) model and the computational approaches are extensively applied for the state assessment in the switching-like pulse electromagnetic (EM) applications. They provide theoretical basis and analysis methods for the effective solutions of the unexpected challenges, such as signal integrity, crosstalk as well as the electromagnetic compatibility (EMC) problems. Generally, the per-unit-length (p.u.l.) distributed parameters, such as the resistance, inductance, conductance, and capacitance are frequency dependent quantities. The time domain solution is the most fundamental and intuitive representation for the MTL equations. As a consequence, the research of the time domain MTL solution to accurately and efficiently represent the frequency dependent parameters is of great theoretical significance and engineering importance. The frequency dependent features of the resistance and the internal inductance are mainly due to the skin and proximity effects. The existing Finite Difference Time Domain (FDTD) mehtod is able to include the skin effect, while the influence of the proximity effect is subject to a further investigation. Preliminaryly, in this dissertation, the FDTD formulations for a two-conductor transmission line (TL) and a MTL are systematically derived for the inclusion of the skin and proximity effects, respectively. They are essentially on the basis of an improved two-conductor TL model and an improved MTL model, which are characterized by the partial resistances and the partial inductances. Compared with the case of a two-conductor TL, the resistance and the internal inductance of a conductor in a MTL system are affected by the adjacent ones in the presence of proximity effects. Hence, the modeling of the resistance and the internal inductance for each conductor becomes a key problem to be solved urgently. To this purpose, an idea of coupling MTL model with the numerical solution of EM field is introduced in this dissertation. The Boundary Element Method (BEM) formulation enforcing high order surface impedance boundary condition (SIBC) is adopted for a coupling solution to the resistance and internal inductance for each conductor (in a MTL and in a two-conductor TL). As far as the terminal representation of a MTL is concerned, the computational efficiency of the FDTD approach needs to be improved. Consequently, the Node-to-node Admittance Functions (NAFs) models are presented for this purpose. During the implementation of the NAFs model, the traditional MTL model can not be applied to extract the terminal admittance matrix (TAM). This is due to the fact that the real parts of the poles for the admittance in the TAM regarding to the reference conductor are greater than zeros after the approximation via a matrix rational approximations (MRAs) technique. This as a matter of fact leads to an unstable NAFs model. Consequently, an MTL model which is characterized by the loop resistances and the loop inductances is established in this dissertation. In this case, the stability of the resulting NAFs model is guaranteed. The voltages of each terminal for the conductors with respect to the reference one can be accurately computed. Particularly, in the MTL model with loop parameters, the reference conductor is treated as an ideal one, so that the voltage across the two ends is zero. It is included into the voltages for the other conductors as a matter of fact. To uniquely compute the voltage across each conductor, an improved NAFs model based on the improved MTL model is proposed for a frequency dependent lossy MTL. The main research activities and highlighted achievements are structured as follows: (1) An improved circuit model for the two-conductor TL which is characterized by the partial resistances and the partial inductances and the relevant analysis approach are presented when the distributions of the voltage and the current at the internal points along a TL are concerned. A frequency selective partial internal impedance model is proposed based on the features of the skin and proximity effects. At low frequency range the partial internal impedance is taken as its dc value; and at high frequency range it is computed via the BEM-SIBC. Subsequently, an explicit expression of the smooth transition model for partial internal impedance is obtained from the frequency selective one with the aid of first order low and high pass filters. The TL equations with convolution scheme is detailed deduced. Subsequently, a FDTD formulation is obtained by the discretizations of the TL equations and the required convolution integral via a central difference approximation. In addition, a recursive convolution technique is adopted for a computational efficient FDTD formulation, which solves the problems caused by the direct convolution, such as the high storage requirement and the lengthy computation. The main error sources and the stability criterions of the proposed FDTD formulation are detailed discussed. Its correctness and effectiveness is validated by comparison with a reference inverse fast Fourier transform (IFFT) solution. In addition, it shows an improvement in terms of the accuracy with respect to a traditional one, which does not consider proximity effect. (2) An improved MTL model with the inclusion of partial resistances and partial inductances is presented, which is an indirect extension for the case of a two-conductor TL. Subsequently, the coresponding analysis mehtod is described. On this basis, the MTL equations in time, in frequency as well as in Laplace domains are reformulated in matrix notations. Specifically, the increased complexities of loop resistance and loop inductance matrices are discussed in the presence of skin and proximity effects. A frequency selective internal impedance model for each conductor is obtained based on the low and high frequency approximations. Further, a solution of the transition model for loop internal impedance with different physical dimensions is investigated. The recursion FDTD formulation with recursive convolution for a frequency dependent lossy MTL is detailed derived. Remarkably, the BEM-SIBC solution is embedded into a developed FDTD program for a high efficient field-circuit coupling scheme. Numerical studies indicate the correctness and practical applicability of the proposed FDTD formulation. (3) In view of the terminal representation, a MTL model including the loop resistances and loop inductances with the confirmed effectiveness is presented. Two approaches are described to extract the targeted TAM for a MTL with the desired length. An explicit relation between the entries of the TAM and the admittance elements of the NAFs is well-established. The TAM is amenable to the MRAs process directly, which yields a stable rational model. Consequently, the rational model for NAFs is deduced from the one for TAM. Subsequently, a NAFs model for a frequency dependent lossy MTL is implemented with the aid of a circuit synthetization method. Thereby making the voltages of the terminals for each conductor with respect to the reference one is computable. To realize a NAFs model for a long MTL, a segmenting method is briefly introduced. Numerical validations demonstrate the correctness and effectiveness of the proposed NAFs model. Interestingly, it can be an efficient substitution for the original MTL model due to the high computational efficiency in frequency domain. Importantly, it shows improvements in terms of the accuracy and the computational efficiency against the proposed FDTD formulation. (4) For a unique computation of the voltage across each conductor, an improved NAFs model is proposed on the basis of the improved MTL circuit model. Remarkably, it overcomes the disadvantage of the FDTD technique and the NAFs model, which can not accurately compute the voltage across each conductor. The direct extraction method is adopted to determine the modified TAM for a MTL with the desired length. Remarkably, the introduction of partial resistances and partial inductances facilitates the modified TAM can be approximated via the MRAs with a further stable rational model. Thus, by using the circuit synthetization approach, the terminal representation for a frequency dependent lossy MTL can be fast and accurately obtained. Although the improved NAFs model is evidently at the price of the increased model complexity in terms of the number of the terminals and the number of the admittance elements, satisfactory accuracy is achieved both in frequency and in time domains. It gives a similar accuracy with the NAFs model when it comes to the computation of loop voltages, and it still preserves a superior accuracy and the computational efficiency with respect to the proposed FDTD formulation. Additionally, it still significantly reduces the computational time regarding to the improved MTL model.