136 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 21, NO. 1, MARCH 2006 A Brushless Exciter Model Incorporating Multiple Rectifier Modes and Preisach’s Hysteresis Theory Dionysios C. Aliprantis, Member, IEEE, Scott D. Sudhoff, Senior Member, IEEE, and Brian T. Kuhn, Member, IEEE Abstract—A brushless excitation system model is set forth that includes an average-value rectifier representation that is valid for all three rectification modes. Furthermore, magnetic hysteresis is incorporated into the -axis of the excitation using Preisach’s theory. The resulting model is very accurate and is ideal for situations where the exciter’s response is of particular interest. The model’s predictions are compared to experimental results. Index Terms—Brushless rotating machines, magnetic hys- teresis, modeling, simulation, synchronous generator excitation, synchronous generators. I. INTRODUCTION RUSHLESS excitation of synchronous generators offers increased reliability and reduced maintenance require- B Fig. 1. Schematic of a brushless synchronous generator. ments [1], [2]. In these systems, both the exciter machine and the rectifier are mounted on the same shaft as the main alternator (Fig. 1). Since the generator’s output voltage is regulated by [14]; it was originally devised for small-signal analyses and controlling the exciter’s field current, the exciter is an integral its applicability to large-disturbance studies remains question- part of a generator’s control loop and has significant impact on able [15]. An average-value machine-rectifier model that allows a power system’s dynamic behavior. linking of a -axes machine model to dc quantities was derived This paper sets forth a brushless exciter model suitable for in [16]. This model is based on the actual physical structure of use in time-domain simulations of power systems. The analysis an electric machine and maintains its validity during large-tran- follows the common approach of decoupling the main generator sient simulations. from the exciter–rectifier. Because of the large inductance of a In this paper, the theory of [16] (which covered only mode I generator’s field winding, the field current is slow varying [3], operation) is extended to all three rectification modes [17]. [4]. Therefore, the modeling problem may be reduced to that This is necessary for brushless excitation systems, because the of a synchronous machine (the exciter) connected to a rectifier exciter’s armature current—directly related to the generator’s load. field current—is strongly linked to power system dynamics For power system studies, detailed waveforms of rotating rec- [3]. During transients, the rectifier’s operation may vary from tifier quantities are usually not important (unless, for example, mode I to the complete short-circuit occurring at the end of diode failures [5] or estimation of winding losses are of in- mode III [6]. The exciter–rectifier configuration is analyzed on terest). Moreover, avoiding the simulation of the internal rec- an average-value basis in a later section. tifier increases computational efficiency and reduces modeling The incorporation of ferromagnetic hysteresis is an additional complexity [6], [7]. The machine-rectifier configuration may be feature of the proposed model. Brushless synchronous gener- viewed as an ac voltage source in series with a constant com- ators may use the exciter’s remanent magnetism to facilitate mutating inductance [8]; however, this overly simplified model self-starting, when no other source is available to power the does not accurately capture the system’s operational character- voltage regulator. However, the magnetization state directly af- istics [9]–[13]. The widely used brushless exciter model pro- fects the level of excitation required to maintain a commanded posed by the IEEE represents the exciter as a first-order system voltage at the generator terminals. Hence, representation of hys- teresis enhances the model’s fidelity with respect to the voltage regulator variables. Manuscript received October 28, 2003; revised September 29, 2004. This work was supported by the “Naval Combat Survivability” effort under Grant Hysteresis is modeled herein using Preisach’s theory [18], N00024-02-NR-60427. Paper no. TEC-00312-2003. [19]. The Preisach model guarantees that minor loops close to D. C. Aliprantis is with the Greek Armed Forces (e-mail: the previous reversal point [20]–[22]. This property is essen- [email protected]). S. D. Sudhoff is with the Department of Electrical and Computer Engi- tial for accurate representation of the exciter’s magnetizing path neering, Purdue University, West Lafayette, IN 47907-1285 USA (e-mail: behavior. Hysteresis models that do not predict closed minor [email protected]). loops, such as the widely used Jiles–Atherton model [23], are B. T. Kuhn is with the SmartSpark Energy Systems, Inc., Champaign, IL 61820 USA (e-mail: [email protected]). not appropriate. To see this, consider a brushless generator con- Digital Object Identifier 10.1109/TEC.2005.847968 nected to a nonlinear load that induces terminal current ripple. 0885-8969/$20.00 © 2005 IEEE Authorized licensed use limited to: Purdue University. Downloaded on August 7, 2009 at 08:08 from IEEE Xplore. Restrictions apply. ALIPRANTIS et al.: A BRUSHLESS EXCITER MODEL INCORPORATING MULTIPLE RECTIFIER MODES 137 Fig. 2. Interconnection block diagram (input–output relationships) for the proposed model. Fig. 3. Illustrations of the elementary magnetic dipole characteristic and the boundary on the Preisach domain. This ripple transfers to the exciter’s magnetizing branch current, and in the “steady-state,” a minor loop trajectory is traced on the The rotating-rectifier average-value model computes the plane. If the loop is not closed, the flux can drift away from average currents flowing in the exciter armature , based on the correct operating point. , the voltage-behind-reactance (VBR) -axis flux linkage This paper begins with a notational and model overview. and the (varying) VBR -axis inductance . (The -axis Next, a brief review of Preisach’s theory is set forth. Then VBR inductance is also used, but is considered constant.) These model development begins in earnest, with the development voltage-behind-reactance quantities are computed from the of the Preisach hysteresis model, a reduced-order machine reduced-order machine model. The hysteresis model performs model, and the rotating-rectifier average-value model. The the computations and bookkeeping required to use Preisach’s paper concludes with a validation of the model by comparison hysteresis theory. Its only input is the -axis magnetizing cur- to experimental results. rent ; its output is the incremental magnetizing inductance that represents the slope of the hysteresis loop at a given II. NOTATION AND MODEL OVERVIEW instant. The integrations of the state equations are performed Throughout this work, matrix and vector quantities appear in inside the reduced-order machine model block. The states are bold font. The primed stator quantities denote referral to the and the -axis magnetizing flux . The aforementioned rotor through the turns ratio, which is defined as the ratio of variables will be defined formally in the ensuing analysis. armature-to-field turns . The electrical rotor Notice that the proposed model is applicable whether hysteresis position and speed are times the mechanical rotor is represented or not; in case of a linear magnetizing path, the position , and speed where is the number of poles. hysteresis block is replaced by a constant inductance term. The analysis takes place in the stator reference frame (since the field winding in the exciter machine is located on the stator). III. HYSTERESIS MODELING USING PREISACH’S THEORY The transformation of rotating to stationary variables is Preisach’s theory of magnetic hysteresis is based on the con- defined by [24] cept of elementary magnetic dipoles (also called hysterons). These simple hysteresis operators may be defined by their “up” (1) and “down” switching values and , respectively (Fig. 3). where1 Equivalently, they may be defined by a mean value and a loop width . The behavior of a ferromagnetic material may be thought to arise from a statistical distribution of hysterons. The func- (2) tion which describes the density of hysterons is known as the Since a neutral connection is not present, . Preisach function. It is defined on and is denoted by The components of the proposed excitation model are shown or , depending on which set of coordinates is used. The in Fig. 2. The exciter model connects to the main alternator Preisach function is zero everywhere except on the shaded do- model through the field voltage and current ; it also main of Fig. 3. To explain the shape of this region, it is first requires . The voltage regulator model provides the voltage noted that . The other constraints originate from the ob- to the exciter’s field winding , and receives the current servation that a finite applied field will fully saturate the . The exciter model is comprised of three separate models, material. Thus, all dipoles must obey . Consid- namely, the rotating-rectifier average-value model, the Preisach eration of saturation in the opposite direction yields hysteresis model, and the reduced-order machine model. These three inequalities lead to the triangular domain depicted in Fig. 3. 1The minus sign in the second row and the apparent interchange of the second The domain is divided into two parts: the upper part and third columns from Park’s transformation (as defined