Stability analysis of linear systems with time-varying delay is investigated. In order to highlight the relations between the variation of the delay and the states, redundant equations are introduced to construct a new modelling of the delay system. New types of Lyapunov-Krasovskii functionals are then proposed allowing to reduce the conservatism of the stability criterion. Delay-dependent stability conditions are then formulated in terms of linear matrix inequalities. Finally, several examples show the effectiveness of the proposed methodology.