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This article abandons the traditional Laplace transform and proposes a new method for studying fractional-order circuits, which is the Loop-By-Loop Progressive Iterative Method(LPIM). Firstly, in order to demonstrate the correctness of LPIM, the fractance circuit, which is a relatively mature and simple form in fractional-order circuits, was chosen as the research object. The output signals of fractance circuit were studied for the first time using Laplace transform and LPIM, respectively. The results showed that the conclusions obtained by LPIM were completely consistent with those obtained by Laplace transform method and existing theories, thus verifying the correctness of LPIM. Then, a brand new Fractional-Order Memory Systems (FMS) model is constructed, and based on this model, LPIM is used for the first time to simulate the output signal of Flux-Controlled Fractional-Order Memory Systems (FFMS) that has not been studied so far. The results show that when a sine signal is used as the excitation signal, the output signal of the FFMS intersects at two points, and the output signal is modulated by the frequency of the excitation signal. Finally, combining existing theories, predict the output commonalities of FMS.