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This paper presents a spectrally weighted balanced truncation (SBT) technique for tightly coupled integrated circuit interconnects, when the interconnect circuit parameters change as a result of statistical variations in the manufacturing process. The salient features of this algorithm are the inclusion of the parameter variation in the RLCK interconnect, the guaranteed passivity of the reduced transfer function, and the availability of provable spectrally weighted error bounds for the reduced-order system. This paper shows that the variational balanced truncation technique produces reduced systems that accurately follow the time- and frequency-domain responses of the original system when variations in the circuit parameters are taken into consideration. Experimental results show that the new variational SBT attains, in average, 30% more accuracy than the variational Krylov-subspace-based model-order reduction techniques.