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This paper addresses stability problems in power systems with loads that exhibit constant-power behavior. Instability may occur in such systems due to the negative incremental impedance of constant-power loads (CPLs). Existing approaches to stabilizing such systems require modification of the source and/or the load control characteristics, or isolating the CPL from the rest of the system by additional active devices, which are difficult to implement and often conflict with other system requirements such as control bandwidth, size, weight, and cost. In this paper, we investigate passive damping as a general method to stabilize power systems with CPL. Using a representative system model consisting of a voltage source, an LC filter, and an ideal CPL, we demonstrate that a CPL system can be stabilized by a simple passive damping circuit added to one of the filter elements. Three different damping methods are considered and analytical models are developed for each method to define damping parameters required for stabilizing the system. Time- and frequency-domain measurements from an experimental system are presented to validate the methods.