A visibility graph transforms time series into graphs, facilitating signal processing by advanced graph data mining algorithms. In this paper, based on the classic limited penetrable visibility graph method, we propose a novel mapping method named circular limited penetrable visibility graph, which replaces the linear visibility line in limited penetrable visibility graph with nonlinear visibility arc for pursuing more flexible and reasonable mapping of time series. Tests on degree distribution and some common network features of the generated graphs from typical time series demonstrate that our circular limited penetrable visibility graph can effectively capture the important features of time series and show higher robust classification performance than the traditional limited penetrable visibility graph in the presence of noise. The experiments on real-world time-series datasets of radio and electroencephalogram signals also suggest that the structural features provided by a circular limited penetrable visibility graph, rather than a limited penetrable visibility graph, are more useful for time-series classification, leading to higher accuracy. This classification performance can be further enhanced through structural feature expansion by adopting subgraph networks. All of these results demonstrate the effectiveness of our circular limited penetrable visibility graph model.