In this paper, we analyze the performance of a single-relay network in which the reliability is provided by means of Random Linear Network Coding (RLNC). We consider a scenario when both source and relay nodes can encode packets. Unlike the traditional approach to relay networks, we introduce a passive relay mode, in which the relay node simply retransmits collected packets in case it cannot decode them. In contrast with the previous studies, we derive a novel theoretical framework for the performance characterization of the considered relay network. We extend our analysis to a more general scenario, in which coding coefficients are generated from non-binary fields. The theoretical results are verified using simulation, for both binary and non-binary fields. It is also shown that the passive relay mode significantly improves the performance compared with the active-only case, offering an up to two-fold gain in terms of the decoding probability. The proposed framework can be used as a building block for the analysis of more complex network topologies.