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The prevailing circuit simulation paradigm heavily relies on extraction to perform circuit simulation. In addition, no linear complexity has been reported by existing circuit simulation approaches. In this work, we develop a circuit simulator that bypasses the step of extraction. It produces an RLC (resistor-inductor-capacitor) representation of the linear network without any numerical computation. Such a representation is inherently passive and stable. In addition, it allows for the simulation of a circuit including both nonlinear devices and the linear network in linear complexity. The proposed circuit simulator rigorously captures the coupling between nonlinear circuits and the linear network. As an added bonus, it possesses electromagnetic-physics based accuracy. Application to very large-scale on-chip circuits involving close to 1 million CMOS transistors and interconnects having hundreds of millions of unknowns has demonstrated the superior performance of the proposed extraction-free circuit simulator.