SUMMARYA composite right/left-handed (CRLH) extended equivalent circuit (EEC) FDTD method with a stability criterion based on a Liapunov discrete energy function is presented and applied to investigate several transient and refractive phenomena occurring at the interface between a CRLH metamaterial (MTM) and a purely right-handed (PRH) structure. This formulation consists in an extension of the equivalent circuit (EC) Yee scheme including a left-handed (LH) series capacitance and shunt inductance in addition to the canonical right-handed (RH) series inductance and shunt capacitance, so as to model general CRLH transmission line (TL) MTMs. This CRLH-EEC FDTD scheme is shown to represent a convenient numerical scheme, with remarkable formulation compactness and high computational efficiency, for the analysis of any type of MTM structure. Moreover, this scheme, coupled with a Y-matrix representation of the EEC Yee cell, is shown to rigorously restore the dispersion/attenuation relation of CRLH MTMs. Transient solutions for the propagation of a modulated Gaussian pulse through a PRH-CRLH interface highlights interesting effects such as space-domain compression, infinite phase velocity modulation and the well-known backward wave effect. Frequency-domain analysis of the fields radiated by dipoles through the Veselago-Pendry lens shows how focusing is affected both qualitatively and quantitatively by the orientation of the dipoles.