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In this paper, a general analysis method for ring resonators is proposed. The method utilizes a characteristic matrix to express the recursive relationship between transfer functions of adjacent-order optical paths, and consequently, the total transfer function. The method provides a new way to express Mason's rule in general. At the same time, it greatly simplifies the expression in the ring-resonator case. The method can be used to synthesize any ring-resonator devices. In addition, as individual optical paths' transfer functions are derived in this process, the formulas can be applied to incoherent interferometric measurements and can characterize the device parameters of fabricated complex ring-resonator circuits.