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This paper presents a systematic electromagnetic-circuit method for efficient simulation of coupled interconnects on dispersive dielectrics in the presence of electromagnetic interference. The dispersive substrate of coupled interconnects is characterized by the multiterm Debye or Lorentz formula. According to the differential equations derived from the equivalent circuit modeling of dispersive media, the full-wave finite-difference time-domain algorithm is utilized to capture accurate field in a dispersive substrate. The effects of incident field, under the quasi-TEM assumption, are then represented by the distributed sources along transmission lines. An efficient algorithm based on the waveform relaxation with transverse partitioning (WR-TP) is employed for fast simulation of coupled transmission lines. A simple and clear derivation of the relaxation sources in WR-TP is described and the compact results are given. The impact of dispersive dielectrics on the incident field, and thus to the transient response of interconnects, is investigated by comparison with conventional techniques. Several numerical examples are given to illustrate the accuracy and efficiency of the proposed method.