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Multi-stage reconnection electromagnetic launcher is not a simple addition of some single-stage launching systems, there is a stage coupling between each driving coils. In this paper, the mathematical models of single and multi-stage reconnection electromagnetic launchers are described in detail, and the inter-stage coupling at different velocities is discussed respectively. The inter-stage coupling coefficient is very small when the velocity is low, and it increases as the velocity increases. [ABSTRACT FROM AUTHOR]