The brushless doubly fed reluctance machine (BDFRM) is the least known of a group of electrical machines that include the classic cascaded induction machine, the brushless doubly-fed induction machine (BDFIM), and the double-fed slip ring induction machine (DFSRIM). Since its initial development some 30 years ago, the BDFRM has been largely ignored because of the performance limitations imposed by the reluctance rotor design. However, improvements in reluctance rotors, a by-product of the development of the synchronous reluctance machine, have resulted in renewed interest in the BDFRM. This together with the promise of higher efficiency and simpler control compared to the BDFIM means that further investigation of the BDFRM is warranted. This paper is designed to be a starting point for research into the BDFRM. It develops the fundamental modelling equations that are required to carry out research into its dynamics and control. The approach is partly tutorial in nature as it presents some "well-known" analysis techniques that are scattered throughout the literature on the machine. It develops from first principles the space vector model of the machine, which is then used to derive the steady-state BDFRM equations. Standard sinusoidal spatial variation and linearity assumptions are used throughout the analysis. Where relevant, the physical concepts behind the machine's operation are emphasized. In addition to the full dynamic model of the machine, the paper also introduces a per-unit system that forms the basis for machine-independent performance expressions.