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A new unconditionally stable three-dimensional (3-D) transmisson-line (TLM) algorithm is presented. It is stable regardless of the selected time-step. This new algorithm is based on a split-step theory, whose numerical implementation is given in detail. In addition, the theoretical proof of its unconditional stability is provided. This feature provides some potential advantage for time-domain electromagnetic-field computation as the number of iterations can be arbitrarily reduced for a given space sampling. Unfortunately, it is shown that the numerical dispersion of the new scheme increases when the time-step is different from the maximum value of the standard TLM. However, it is shown that some substantial computer cost reduction can be achieved when irregular meshing is used, as compared to classical 3-D TLM schemes. Thus, a new meshing strategy to improve the scheme accuracy is presented and validated through several examples.