IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 6, JUNE 2011 2313 Generalized Vector Model for the Brushless Doubly-Fed Machine With a Nested-Loop Rotor Farhad Barati, Shiyi Shao, Ehsan Abdi, Member, IEEE, Hashem Oraee, and Richard McMahon Abstract—This paper presents a generic vector model for the brushless doubly-fed machine (BDFM) with a nested-loop rotor. The vector model is presented for a generic p 1 /p 2 pole-pair BDFM which may have any number of loops per nest. The vector transformations utilized in the derivation of the vector model are provided in general form such that the model can be represented in any appropriate reference frames according to the specific application of the model, such as BDFM analysis, simulations, or control system synthesis. The derived vector model benefits from the feature of reduced order in comparison with the coupled- circuit model, which is the most detailed model for the BDFM, while maintaining the accuracy by including the effects of all loops in each nest. The vector model is developed in MATLAB/Simulink based on the rotor reference frames for a 180-frame-size prototype BDFM, and its predictions for the machine performance have been compared with those of the coupled-circuit model as well as verified experimentally. Index Terms—Brushless doubly-fed machine (BDFM), coupled- circuit model, nested-loop rotor, rotor reference frames, vector model, vector transformations. NOMENCLATURE v s1 , v s2 , v r Voltage matrix for stator1, stator2, and rotor. i s1 , i s2 , i r Current matrix for stator1, stator2, and rotor. λ s1 , λ s2 , λ r Flux matrix for stator1, stator2, and rotor. R s1 , R s2 , R r Resistance matrix for stator1, stator2, and rotor. M s1 , M s2 , M r Inductance matrix for stator1, stator2, and rotor. M s1r , M s2r Mutual inductance matrix for stator1-rotor and stator2-rotor. J, B Rotor moment of inertia and friction coefficient. p 1 ,p 2 Pole pairs of stator1 and stator2 windings. θ s1 ,θ s2 ,ϕ r Arbitrary functions of time in stator1, stator2, and rotor transformations. ω s1 ,ω s2 Time-derivatives of θ s1 ,θ s2 . Manuscript received March 14, 2010; revised June 8, 2010; accepted June 28, 2010. Date of publication August 5, 2010; date of current version May 13, 2011. F. Barati and H. Oraee are with the Department of Electrical Engi- neering, Sharif University of Technology, Tehran 11365-8639, Iran (e-mail: barati@ee.sharif.edu; oraee@sharif.edu). S. Shao, E. Abdi and R. McMahon are with the Department of En- gineering, University of Cambridge, CB3 0FA Cambridge, U.K., and also with Wind Technologies Ltd., St Johns Innovation Centre, CB4 0WS Cambridge, U.K. (e-mail: ss656@cam.ac.uk; shiyi.shao@windtechs.com; ea257@cam.ac.uk; ehsan.abdi@windtechs.com; ram1@cam.ac.uk; richard. mcmahon@windtechs.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2010.2064279 Fig. 1. Nested-loop rotor for the prototype BDFM. V s1 , I s1 , Λ s1 Stator1 voltage, current, and flux vectors. V s2 , I s2 , Λ s2 Stator2 voltage, current, and flux vectors. V r , I r , Λ r Rotor voltage, current, and flux vectors. real{} Real part of a complex number. Z ∗ ,Z t Complex conjugate and transpose of Z . I. I NTRODUCTION T HE brushless doubly fed machine (BDFM) is an attractive alternative to the doubly-fed induction generator (DFIG) in wind power applications [1], [2]. It also has the potential to be employed as a motor in variable-speed drive applications [3], [4]. The BDFM comprises two electrically independent non- mutually coupled balanced three-phase windings on the same stator core and a special rotor which couples to both fields of stator windings. The nested-loop type of rotor is the most well known [2] and consists of nests which are equally spaced around the circumference. The number of nests is equal to the sum of the stator windings’ pole-pairs [1]. Fig. 1 shows the nested-loop rotor for the authors’ BDFM. This machine has four-pole and eight-pole stator windings and, hence, there are six nests, of three loops each, in the rotor. A common end ring shorts all the loops at one end of the rotor. The BDFM is not stable over the operating speed range and, therefore, a controller is required to stabilize the machine as well as to meet other dynamic and steady-state performance requirements [5]. Vector control methods, also known as field- oriented control, have been applied to the BDFM and shown to give good performance [6]–[11]. In order to understand the requirements for a suitable controller and to determine the sta- bility margins, an appropriate mathematical model is required to describe the machine’s dynamic performance; this model then forms the basis for the controller design and analysis. 0278-0046/$26.00 © 2010 IEEE