Various magnetically coupled impedance source (MCIS) networks have been proposed in the literature for increasing voltage gain and modulation index simultaneously while reducing the number of passive components used in the converter. However, applications of such networks have been limited by their discontinuous currents drawn from the sources and/or high stresses experienced by their components. This paper thus proposes three new MCIS networks named, respectively, as quasi-Y-source, quasi-Γ-Z-source, and quasi-T-source or quasi-trans-Z-source networks. These new networks inherit all advantages of the existing MCIS networks. In addition, they demonstrate advantages like continuous input currents, reduced source stresses, and lower component ratings that are not achievable by other existing networks. Further, dc-current-blocking capacitors used in the networks help to avoid saturation of magnetic core. Derivations of all two-winding MCIS networks, including existing and new networks, from the generalized three-winding MCIS networks are then systematically illustrated, before comparing them. Operational principles, mathematical derivations, simulation, and experimental results of all studied networks have been presented in the paper.