296 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 27, NO. 2, JUNE 2012 Dynamic Average-Value Modeling of 120 ◦ VSI-Commutated Brushless DC Motors With Trapezoidal Back EMF Kamran Tabarraee, Member, IEEE, Jaishankar Iyer, Student Member, IEEE, Hamid Atighechi, Student Member, IEEE, and Juri Jatskevich, Senior Member, IEEE Abstract—The 120 ◦ voltage source inverter driven brushless dc (BLDC) motors are very common in many applications. This pa- per extends the previous work and presents an improved dynamic average-value model for such BLDC motor-drive systems. The new model is explicit and uses a proper qd model of the permanent magnet synchronous machine with nonsinusoidal rotor flux. The model utilizes multiple reference frame theory to properly include the back EMF harmonics as well as commutation and conduction intervals into the averaged voltage and torque relationships. The commutation angle is readily obtained from the detailed simula- tion. The proposed model is demonstrated on a typical industrial BLDC motor with trapezoidal back EMF waveforms. The results of studies are compared with experimental measurements as well as previously established models, whereas the new model is shown to provide appreciable improvement. Index Terms—Average-value model (AVM), brushless dc (BLDC) motor, trapezoidal back EMF. I. INTRODUCTION A BRUSHLESS dc (BLDC) motor–inverter system consists of a permanent magnet synchronous machine (PMSM) that is driven by a voltage source inverter (VSI). Typically such motors provide good torque–speed characteristics, fast dynamic response, high efficiency, long life, etc., which make them fa- vorable in wide range of applications including industrial au- tomation, instrumentation, and many other equipment and servo applications. This paper considers typical VSI-driven BLDC motors wherein the inverter operates using 120 ◦ commutation method [1]–[3]. In this switching logic, each phase is allowed to be open circuited for a fraction of revolution, giving rise to complicated commutation–conduction patterns of the stator cur- rents [1]. Steady-state analysis of such motors has been carried out by several researchers [4]–[6]. Manuscript received June 9, 2011; revised November 23, 2011; accepted January 31, 2012. Date of publication March 19, 2012; date of current version May 18, 2012. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under the Discovery Grant. Paper no. TEC-00276-2011 K. Tabarraee was with the Department of Electrical and Computer Engi- neering, University of British Columbia, Vancouver, BC V6T1Z4, Canada. He is now with Powertech Labs, Inc., Surrey, BC V3W 3J4, Canada (e-mail: Kamran.Tabarraee@powertechlabs.com). J. Iyer, H. Atighechi, and J. Jatskevich are with the Department of Elec- trical and Computer Engineering, University of British Columbia, Vancou- ver, BC V6T1Z4, Canada (e-mail: jaiiyer@ece.ubc.ca; hatighechi@ece.ubc.ca; jurij@ece.ubc.ca). 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/TEC.2012.2188032 The detailed modeling of such BLDC motor–inverter system has been described in literature quite well [1], [2], [4]–[8] and can be easily carried out using a number of simulation pack- ages [9]–[13]. In many available detailed models, it is often assumed that the induced back EMF waveform of the machine is sinusoidal [1], [3], [6], [14]. However, the actual back EMF waveform might be quite nonsinusoidal. Including the back EMF harmonics into the voltage and torque equations increases the accuracy of the model. For system-level transient studies or small-signal analysis of electromechanical systems with BLDC motor drives (such as aircraft, vehicular, or industrial automation systems), the so- called average-value models (AVMs) are indispensable. The AVMs approximate the slower system dynamics while neglect- ing the details of fast switching, which is removed from the model by the means of averaging. Because of that, the AVMs can typically use much larger integration time step leading to faster simulation times. The AVMs are also continuous and can be linearized about a desired operating point for small-signal analysis that is readily available within many simulation pack- ages, e.g., [12], [13]. The average-value modeling approach has been recognized by many researchers as a powerful tool to study and analyze complicated power-electronic-based systems when the dynamics of interest are slower than the switching frequency. Interested reader can find a summary of promising approaches and their application to the modeling and analysis of systems transients (including the vehicular power systems) in recent reports by the IEEE Task Force on Dynamic Average Modeling [15], [16]. In general, derivation of dynamic AVMs requires careful av- eraging of the stator phase voltages and currents over a prototyp- ical switching interval (SI) to find the corresponding average- value relationships for the state variables and electromagnetic torque. A pioneering work in this direction has been the AVM for the BLDC motor–inverter system with sinusoidal back EMF [3]. This approach has been extended to include both conduction and commutation subintervals [14]. Other AVMs developed for the nonsinusoidal back EMF PMSM include [17] and [18], both of which assume operation in continuous-current mode. However, the average-value modeling of the 120 ◦ VSI-driven BLDC mo- tors with trapezoidal back EMF becomes more challenging due to the discontinuous current and harmonics in the voltage and torque equations, and to the best of our knowledge, has not been addressed in the literature. 0885-8969/$31.00 © 2012 IEEE