Recently Lagrangian dynamics has been applied to transforming integrated magnetic components into equivalent circuits of basic magnetic components such as transformers and inductors. Although the method is beneficial in simple and systematic derivation in many cases, it sometimes leads to different circuits from those by conventional methods. Hence, the Lagrangian method needs equivalence evaluation of transformation. As a case study, this paper derives equivalent circuits of the integrated winding coupled inductor using the Lagrangian method and a conventional method. The equivalent circuits are investigated to verify their consistency with magnetic circuit model and experimental behavior. As a result, the Lagrangian method yields a simpler circuit than that by the conventional method. Nonetheless, both circuits are found functionally equivalent to the magnetic circuit model and consistent with the experiment. The result suggests that the Lagrangian method provides proper transformation and is useful for deriving simple equivalent circuits.