A Field Reconstruction Method for Optimal ... - Purdue Engineering

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 21, NO. 2, JUNE 2006

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A Field Reconstruction Method for Optimal Excitation of Permanent Magnet Synchronous Machines Weidong Zhu, Babak Fahimi, Senior Member, IEEE, and Steve Pekarek, Member, IEEE

Abstract—Vibration caused by torque ripple and radial force harmonics is a concern in many applications of permanent magnet synchronous machines (PMSMs). Alternative methods of machine design and/or stator excitation to minimize torque ripple have received considerable attention in recent years. Comparatively, methods to minimize radial force harmonics have received less attention. In this paper, a field reconstruction (FR) method is derived that provides a designer with the capability to rapidly determine the radial and tangential components of force under arbitrary stator excitation. Using the field reconstruction method, stator current waveforms that minimize the ripple of both torque and radial force are derived subject to the constraint of maintaining a satisfactory level of torque density. Index Terms—Finite element analysis (FEA), force density, permanent magnet machine, torque ripple.

I. INTRODUCTION ERMANENT magnet synchronous machines (PMSMs) are widely used in high-performance applications due to their relatively high-power density, absence of brushes, negligible rotor losses, high efficiency, and ease of control. However, the acoustic noise and vibration originated by torqueand radial-force harmonics remains a concern in many applications. As a result, the study of electromagnetic forces in PMSMs has attracted considerable attention over the past decade [1]–[12]. The electromagnetic torque of a PMSM is created by the tangential component of the force acting on the rotor core, which in turn is determined by the flux distribution in the airgap. Therefore, the torque distribution is affected by any factor that has influence on the magnetic field in the airgap, including the method of excitation and machine geometry. Accordingly, techniques to minimize torque ripple are classified into two major categories. The first includes machine design approaches that effectively attempt to limit non-idealities and/or compensate for them. These include skewing of the ro-

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Manuscript received January 26, 2005; revised May 5, 2005. This work was supported in part by the office of Naval Research under Grant N00014-04-10815 and Grant N00014-02-1-0623. Paper no. TEC-00021-2005. W. Zhu is with the Department of Electrical and Computer Engineering, University of Missouri-Rolla, Rolla, MO 65409 USA (e-mail: [email protected]). B. Fahimi is with Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX 76019 USA (e-mail: [email protected]). S. Pekarek is with the Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TEC.2005.859979

tor or stator, using fractional slot pitch windings, and introducing dummy slots [1], [2]. The second category includes techniques to modify the stator excitation of the machine to obtain a smoother torque waveform [1], [3]–[12]. Though the techniques in the first category are among the most effective ways for torque ripple minimization, they may not be sufficient or appropriate in some applications. Specifically, skewing and fractional-slot will result in reduction of average torque, while using dummy slots does not necessarily reduce the torque ripple but increases the frequency of fluctuation, so its effect depends upon the mechanical system. In addition, some design methods require high-precision manufacturing, which adds cost. Thus, in certain circumstances, current profiling represents a preferred solution. Furthermore, new advancements in power electronics and digital signal processing-based controllers have paved the way for effective profiling of the stator excitation [13]. Radial attraction forces are viewed as another source of vibration and acoustic noise in PMSMs. However, in contrast to torque-ripple mitigation, there has been relatively little effort on the development of design or excitation strategies to mitigate the effects of normal harmonics— other than to alter the natural frequencies of the stator structure. In theory, the tools required to optimize a machine for minimal torque ripple or radial force ripple are established and commercially available. Specifically, using a Maxwell stress tensor (MST) method coupled with a finite element analysis (FEA)based solution for the fields, the tangential and radial forces can be evaluated as a function of phase current excitation. The optimal phase current waveforms could then be obtained using one of a number of optimization algorithms. However, in practice finite element computations are time consuming and thus effectively narrow the domain that can be explored for optimal solutions. In this paper, a field reconstruction (FR) method is introduced to expedite the process of searching for optimal current profiles for PMSMs. The FR method utilizes the fields created by current in a single slot, along with fields created by the permanent magnet of the rotor over a single stator tooth pitch, to construct the entire force profile for the machine under arbitrary stator excitation and rotor position. The method enables a designer to generate the tangential and radial components of the force in a more timely fashion. Its usefulness and accuracy is demonstrated in a design example where the objective is to minimize the harmonics of both normal and tangential components

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to MST, the tangential and normal force densities in the airgap can be expressed as ft = Bn Bt /µ0 fn = Bn2 − Bt2 2µ0

(1) (2)

where Bn and Bt are the radial and tangential components of the magnetic flux density, respectively, and µ0 is the permeability of air. Using (1) and (2), the tangential force (per axial length) can be obtained using line integration (3) ft · dl Ft = Γ

Fig. 1.

Cross section view of the PM machine and coordinate system.

of force, subject to the constraint of bounded stator current amplitudes. II. MACHINE MODEL The PMSM studied in this research is shown in Fig. 1. It is a 1-hp, 2000-rpm, 3-phase, 4-pole, 12-slot, surface-mounted machine that is used in home appliance applications. In Fig. 1, the phase-a winding magnetic axis and the q-axis of the rotor are shown, respectively. The mechanical rotor position (θrm ) is defined as the angle between the two axes. For the purpose of analysis, the angle φsm is defined as the position on the stator relative to the midpoint of the phase-a stator slot shown and the angle φrm is defined as the position on the rotor relative to the midpoint of the PM shown. Although not shown on the diagram, the separation between angles φrm and φsm is the same as the angle between the q-and as-axis (θrm ). So-called electrical angles φs , φr , and θr are defined by multiplying the mechanical angles by the number of pole pairs. The following assumptions have been made for the analysis provided. • The stator teeth and permanent magnets are rigid; no deformation due to radial and tangential force is experienced by these components. • The stator windings are concentrated. They are wound at a full-pitch. • The permanent magnets are not demagnetized by the flux introduced by the phase currents. • The flux density in the axial direction (z-axis) is assumed zero (no end effects). • Hysteresis and eddy currents are neglected. III. FORCE CALCULATION Alternative methods for calculation of electromagnetic forces in electrical machines are summarized in [14]. Among them the MST method is a popular approach due to its simplicity and provision to supply detailed local force distribution. According

where Γ is the integration contour in the airgap. Here, we define a similar integral to represent the radial force over an electrical cycle. Specifically 2π fn rdφs (4) Fn = 0

where r is the radius of the integration contour. The normal force defined in (4) is the integration of stress. In practice Bn is larger than Bt and therefore the stress is always positive (i.e., force density is in a direction from rotor to stator over the entire contour). Therefore, (4) is a number (greater than zero) that is representative of the size of the radial forces. Its nonzero value is not an indication that there is a net force acting to push/pull the stator and rotor. Indeed, for a balanced excitation and uniform stator/rotor structure the net vector normal force acting on the rotor is zero. The MST method provides great simplicity and reasonable accuracy [15] to calculate the local force distribution, though it relies upon the accuracy of the local field values from an FEA solution. In this research, the normal and tangential components of flux density and force were evaluated using a FEA model created using the commercial package Maxwell [16]. A meshing with 21,934 triangles was used in the force calculation. To minimize error, the contour of integration was established in the middle of the airgap [15]. Hence the distribution of the force components is calculated in the middle of the airgap. A convergence test was conducted to insure the accuracy. Specifically, the number of triangles was increased to roughly 130, 000 and the difference between the forces calculated was within 0.5%. For a PM machine, the phase windings and PMs generate the magnetic field in the airgap. As a basis for the FR technique, it is assumed that the iron operates in a linear magnetic region. This is justified by the fact that a surface-mounted PMSM usually has a relatively large airgap and saturation does not have a significant effect under nominal operation. The assumption is found to be reasonable through comparison with the FEA model. Specifically, in the FEA model a nonlinear B-H curve is used for both stator and rotor steel. Based upon the assumption of linearity, the superposition rule is applied and the normal and tangential flux density in the airgap are expressed as Bn = Bnpm + Bns

(5)

Bt = Btpm + Bts

(6)

ZHU et al.: A FIELD RECONSTRUCTION METHOD FOR OPTIMAL EXCITATION OF PERMANENT MAGNET SYNCHRONOUS MACHINES

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Fig. 3. An optimization routine in which FEA is used for determining optimized stator excitation.

Fig. 2. Normal and tangential components of flux density due to current in a single stator slot.

machine assuming stator currents embedded in stator iron and 2) a machine assuming stator currents in the airgap with a smooth stator inner surface. In both cases, the results indicate a solution with even and odd symmetry, respectively. Substituting the expressions given in (5)–(8) into (1) and (2), one can compute the tangential and normal components of the force density as given by L L 1 Btsk + Btpm · Bnpm + Bnsk (11) ft = µ0 k =1 k =1  2 L 2  L 1  fn = Bnpm + Bnsk − Btsk + Btpm  . 2µ0 k =1

where Bnpm , Btpm , Bns , and Bts denote the normal and tangential components of the flux density created by the PM and stator excitation, respectively. Neglecting the effect of end windings, the resultant magnetic fields created by the stator windings are the sum of the individual fields created by the current in each stator slot. Specifically, if Bnsk and Btsk represent the normal and tangential flux densities created by current in kth slot, then the overall normal and tangential components of the magnetic flux from stator windings are Bns =

L

Bnsk

(7)

Btsk

(8)

k =1

Bts =

L k =1

where L is the number of stator slots. To evaluate the functions (7) and (8), the local flux densities created by current in the k th slot are expressed as a generalized function that is proportional to the current. Specifically we define Btsk (φs ) = If1 (φs )

(9)

Bnsk (φs ) = If2 (φs )

(10)

where f1 and f2 are associated with the geometry of the machine only. An FEA solution is used to establish the properties of Btsk (φs ) and Bnsk (φs ). The result of the FEA solution for the given machine is shown in Fig. 2. From this figure it can be seen that f1 and f2 are even and odd functions of φs respectively. This is consistent with the analytical derivations made on simplified machines. Specifically in [17], solutions of the Poisson equation in cylindrical coordinates are obtained for 1) a

k =1

(12) IV. FIELD RECONSTRUCTION FEA is an effective approach in the analysis and synthesis of electric machines. In theory, it can be used along with an optimization routine to obtain an optimal phase current excitation pattern to minimize ripple in tangential and normal components of force. An example of how such a strategy could be implemented is shown in Fig. 3. As shown, an FEA model obtains a current excitation calculated from an optimization routine. The forces calculated using the FEA routine are then used to evaluate a cost function, and the result is provided back to the optimization routine to adjust the excitation strategy based on the feedback. This iteration process is terminated as a user-defined objective is achieved. In practice, due to the complex geometry and existence of multiple components with different magnetic characteristic, the implementation of the FEA in electrical machine design usually comes with an inherent computational cost. The iteration process encompassed in the optimization makes the computational load more intensive. An FR approach is introduced to reduce the computational requirements of design optimization. The overall process follows the following routine. First, at an arbitrary rotor position, a single FEA evaluation is used to obtain the normal and tangential components of the magnetic flux density due to current in a single stator slot. From this evaluation, the functions f1 (φs ) and f2 (φs ) in (9) and (10) are computed. Using these functions as a basis, the magnetic flux density due to the overall stator excitation can be obtained. Specifically, if the flux density components from a current of I0 sitting in the slot 0th with position at φs0

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Fig. 4.


Field reconstruction process. Left: Normal flux density reconstruction. Right: Tangential flux density reconstruction.


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Fig. 6. Control optimization implementation using field reconstruction approach.

Fig. 5.

Flow chart of field reconstruction process.

are Bns0 and Bts0 , respectively, then the components created by current I in the k th slot can be expressed as Bnsk =

I Bns0 (φs − kγ) I0

(13)

Btsk =

I Bts0 (φs − kγ) I0

(14)

where γ is the slot pitch. Using the value of current in each slot, the flux densities created by the stator windings can be reconstructed using (7), (8), (13), and (14). In the second step, the magnetic flux densities created by the rotor are determined using FEA evaluations at several rotor positions over a stator slot pitch (with the stator de-energized). These evaluations are used to obtain the values of Bnpm and Btpm in (5) and (6). An example of the field reconstruction process at a single rotor position where Ias−peak = 4.6 A, Ibs = Ics = − I2as , is shown in Fig. 4. A flow-chart showing the overall reconstruction process is given in Fig. 5. The coupling of the FR-based model and an optimization routine is shown in Fig. 6. From Fig. 6, it can be seen that the objective functions are evaluated using the reconstructed magnetic flux densities, which requires minimal computational effort. Figs. 7 and 8 illustrate the comparison of the tangential and normal flux densities obtained from reconstruction and directly from FEA under the same excitation. The greatest difference between the two field quantities are 0.02 T for the tangential and 0.03 T for the normal component, which are 7% and 5% of peak values respectively. However, it can be seen that most of the details are provided in the reconstructed waveform.

Fig. 7. Reconstructed and FEA results of tangential flux densities in airgap. (a) Reconstructed and (b) FEA results.

V. RESULTS A. Example I A first example of using the FR method to generate ripplefree normal and tangential forces is considered in this section. It is assumed the rotor contains parallel-magnetized magnets. The

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Fig. 9. “Optimal” phase currents for ripple-free radial force and torque— obtained using the FR process coupled to optimization algorithm.

Fig. 8. Reconstructed and FEA results of normal flux densities in airgap. (a) Reconstructed and (b) FEA results.

overall objective is expressed as Minimize (Ft − Ft )2 + (Fn − Fn )2 Subject to ias + ibs + ics = 0

and

|ias |, |ibs |, |ics | ≤ irated where Ft and Fn represent the average (ripple-free) components of tangential and normal force/axial-length, respectively. The values of Ft = 1000 N/m and Fn = 6200 N/m were set to the values obtained by applying rated sinusoidal stator current (4.6 A) to the machine. The constraint condition is established by the physical connection of the windings (wye-connection) and sourcing capability of the power electronics. Again, it is noted that the normal component of the force is not to be interpreted that a net force is acting to push/pull the stator/rotor structures. Moreover, reduction of the normal force ripple represents a reduction of the integral of the force density and not localized normal force. The objective function is difficult to solve analytically due to the complexity of flux density distribution and the inclu-

sion of integration. In this study, it was solved using a sequential quadratic programming method that is part of the commercial Matlab optimization toolbox [18]. Specifically, the Matlab function fmincon was applied. The inputs to fmincon include the function to be minimized, an initial guess vector x0 = [ias0 ibs0 ics0 ]T , upper and lower bounds on x = [ias ibs ics ]T , along with the constraints expressed that are expressed in matrix form [1 1 1]x = 0. Expressions for Ft (x), Fn (x) are derived using (13) and (14) to solve for (11) and (12) and numerically integrating the result to obtain (3) and (4). In the Matlab implementation of the sequential programming method, the line search is performed using a merit function approach [18]. Approximately 40 iterations were required to obtain optimal excitation at each rotor position. The optimal phase currents obtained are shown in Fig. 9. The waveform of phase currents is similar to a trapezoidal shape, indicating a nonsinusoidal excitation for ripple-free tangential and normal force generation. For validation, the force distribution under the optimal phase currents was obtained using FEA. Figs. 10 and 11 show the radial and tangential force as well as the spectrum and comparison between the machine energized with the optimized stator current and sinusoidal currents respectively. The average values of tangential and normal force predicted by FEA are 1100 and 6300 N/m, respectively, which are slightly different from those obtained from the FR method. However, it can be seen that both tangential and normal force ripple have been reduced significantly. Specifically, the sixth harmonic of tangential and normal force are reduced from 73 and 86 N/m in the sinusoidal excitation case, to only 1 and 6 N/m, respectively, with optimal phase current pattern. B. Example II To demonstrate the versatility of the method, it is useful to briefly consider a second example. For this example, the rotor of the machine used in the initial study was assumed to have


Fig. 10. Comparison of the radial force between the “optimal” stator current and sinusoidal phase currents—forces calculated using FEA model.

Fig. 11. Comparison of the tangential force between the “optimal” stator current and sinusoidal phase currents—forces calculated using FEA model.

radial-magnetized magnets. Fig. 12 shows the waveform of the phase current obtained from the optimization. The normal and tangential forces as well as their spectrum are shown in Figs. 13 and 14. Since the PM of the machine is radial-magnetized, it becomes more reasonable to compare the force results to a trapezoidal excitation, which was used to provide a baseline study. As seen in the initial study, both the ripples of normal and tangential forces are reduced substantially. Specifically, the sixth, 12th, and 18th harmonic of tangential force has a drop from 26.4, 24.42, and 9.58 N/m to 0.3, 1.57, and 0.55 N/m, respectively. It can be seen that the “optimized” stator waveforms are very similar to that of parallel-magnetized PM machine in terms of the length of transition and the so-called “flat” stages. However, the waveform does have distinct differences. In fact, a careful inspection of the two set of waveforms indicates an increase of about 8% in the ripple along with changes in the rising profile.

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Fig. 12. “Optimal” phase currents for ripple-free normal force and torque for the radial-magnetized version of the machine—obtained using the FR process coupled to optimization routine.

Fig. 13. Comparison of the normal force between the “optimal” stator current and trapezoidal phase currents for the radial-magnetized version of machine— force calculated using FEA model.

VI. DISCUSSION While the above examples demonstrate the power of the FR method in minimizing tangential and normal force ripple, this method can be applied in other design and control scenarios by modifying objective and constraint conditions. As an example, one could control only the normal harmonics, or attempt to minimize the rms current subject to maintaining average force with minimal ripple of torque and normal force. Thus this method is a powerful tool in considering optimal designs and controls of PMSMs. To consider the impact of the approach on a designer’s ability to search for “optimal” designs, it is helpful to compare and contrast the approach to existing techniques. The computation time for a single FEA evaluation of the machine studied is roughly 5 min using a 1.6-Gb Pentium 4 machine (based upon a meshing density of ∼21 000 elements). The number of iterations of the optimization algorithm used for

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Fig. 14. Comparison of the tangential force between the “optimal” stator current and trapezoidal phase currents for the radial-magnetized version of machine—force calculated using FEA model.


is clear from Figs. 2 and 4 that this cannot be assumed in general, and certainly cannot be assumed for the machine studied. Therefore, although more time-consuming than analytical models, the primary attribute of FR method is the ability to predict the magnetic fields, and consequently the normal and tangential force distribution, at any given point within the airgap for arbitrary rotor/stator geometry (and excitation) with an accuracy which is comparable to that of FE analysis. Considering the accuracy of the MST approach using FEA, the method proposed in [28] was implemented. In this method, the aspect ratio of the triangular elements was virtually kept at 1. This means that the airgap elements are mostly equilateral elements. Moreover, the integrating contour did not pass through any of the vertices. To validate the effectiveness of our approach two alternative contours located at close distances to rotor and stator were chosen. Our results indicated that introducing drastic change in the location of contour constitutes a maximum error of up to 2%, which is an acceptable sensitivity for our design purposes. VII. CONCLUSION

the field reconstruction at each rotor position is (on average) 40. Although the numerical method used for FEA-based optimization would likely be different than used for field-reconstruction, it is assumed the number of iterations would be similar. Neglecting the computation time for the optimization algorithm itself, the computation time required to establish the optimal current profile at a single rotor position using FEA would be approximately 5 × 40 = 200 min. Assuming an FEA solution at 15 rotor positions over a stator slot pitch, then the total computation time to achieve the optimal current using FEA would be approximately 200 × 15 = 3000 min. Using the reconstruction method, the total computation time of the initial FEA is on the order of (15 + 1) × 5 = 80 min. Neglecting the computation time for the optimization algorithm itself, this gives an estimate of a 37:1 computational time advantage between the FEA and reconstruction approach. This represents a conservative estimate in that, in contrast to the FEA model, the FR model has closed form expressions for the forces, which enables one to apply optimization approaches that require derivatives. In contrast, optimization using an FEA model would most likely require use of techniques that require no derivations, i.e., genetic algorithms as an example. Arguably these approaches are more time consuming. There are a host of analytical models that have been used to represent the field and force densities in the airgap of surface mounted pm machines [19]–[26] and are used for machine design. Although these undoubtedly have an advantage over the FR approach in terms of computational effort (since they do not require any FEA computation), the field reconstruction approach has the advantage that there are no assumptions (other than the common assumption of neglecting saturation) made in regard to stator/rotor geometry, magnet thickness, and/or excitation. A relatively common assumption is that the tangential component of flux density introduced by the stator can be neglected. Although this may be accurate for some machines, it

A FR method is introduced. In this method, the normal and tangential components of the magnetic flux density and force are established for arbitrary stator excitation based upon 1) a single FEA evaluation of the magnetic flux density due to current in a single stator slot and 2) an FEA evaluation of the magnetic flux density produced by the rotor rotated over a stator slot pitch (with the stator windings deenergized). The method provides an efficient means of evaluating alternative (and searching for optimal) excitation profiles. As an example, it has been used to establish excitation to achieve an average torque subject to minimal ripple in both the torque and normal force. The results, validated using a full FEA solution, demonstrate the effectiveness. REFERENCES [1] T. M. Jahns and W. L. Soong, “Pulsating torque minimization techniques for permanent magnet ac motor drives-a review,” IEEE Trans. Ind. Electron., vol. 43, no. 2, pp. 321–330, Apr. 1996. [2] D. C. Hanselman, “Effect of skew, pole count and slot count on brushless motor radial force, cogging torque and back EMF,” Inst. Elect. Eng. Proc.—Elect. Power Appl., vol. 144, no. 5, pp. 325–330, Sep. 1997. [3] D. C. Hanselman, “Minimum torque ripple, maximum efficiency excitation of brushless permanent magnet motors,” IEEE Trans. Ind. Electron., vol. 41, no. 3, pp. 292–300, Jun. 1994. [4] E. Favre, L. Cardoletti, and M. Jufer, “Permanent-magnet synchronous motors: A comprehensive approach to cogging torque suppression,” IEEE Trans. Ind. Appl., vol. 29, no. 6, pp. 1141–1149, Nov.–Dec. 1993. [5] P. L. Chapman, S. D. Sudhoff, and C. A. Whitcomb, “Optimal current control strategies for surface-mounted permanent-magnet synchronous machine drives,” IEEE Trans. Energy Convers., vol. 14, no. 4, pp. 1043– 1050, Dec. 1999. [6] F. Colamartino, C. Marchand, and A. Razek, “Torque ripple minimization in a permanent magnet synchronous servodrive,” IEEE Trans. Energy Convers., vol. 14, no. 3, pp. 616–621, Sep. 1999. [7] T. Liu, I. Husain, and M. Elbuluk, “Torque ripple minimization with online parameter estimation using neural networks in permanent magnet synchronous motors,” in Proc. IEEE Ind. Appl. Conf., Oct. 1998, pp. 35– 40. [8] O. Bogosyan, M. Gokasan, and A. Sabanovic, “Robust-adaptive linearization with torque ripple minimization for a pmsm driven single link arm,”


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Weidong Zhu received the B.S. and M.S. degrees in electrical engineering from Shanghai Jiao-tong University, Shanghai, China, in 1995 and 1998, respectively. He is currently working towards the Ph.D. degree in electrical engineering at the University of Missouri-Rolla. From 1998 to 2001, he served as an electrical engineer for ABB China at Shanghai Engineering Center. His interests include analysis and design of power electronics-based electromechanical systems.

Babak Fahimi (S’96–M’00–SM’02) received the Ph.D. degree in electrical engineering from Texas A&M University, College Station, in 1999. Currently, he is an Assistant Professor in the Department of Electrical Engineering, University of Texas-Arlington. His areas of interest include microscopic analysis of electromechanical energy conversion, digital control of adjustable speed motor drives, and design and development of power electronic converters. Dr. Fahimi has been the recipient of IEEE Richard M. Bass power electronics young investigator award as well as the Office of Naval Research young investigator award.

Steve Pekarek (S’89–M’96) received the Ph.D. degree in electrical engineering from Purdue University, West Lafayette, IN, in 1996. From 1997 to 2004, he was an Assistant Professor of Electrical and Computer Engineering at the University of Missouri-Rolla (UMR). He is presently an Associate Professor of Electrical and Computer Engineering at Purdue University and is a Co-Director of the Energy Systems Analysis Consortium. As a faculty member, he has been the Principal Investigator on a number of successful research programs including projects for the Navy, Airforce, Ford Motor Co., Motorola, and Delphi Automotive Systems. The primary focus of these investigations has been the analysis and design of power electronic based architectures for finite inertia power and propulsion systems. Dr. Pekarek is an active member of the IEEE Power Engineering Society, the Society of Automotive Engineers, and the IEEE Power Electronics Society.