446 IEEE Transactions on Energy Conversion, Vol. 10, No. 3, September 1995

Direct Modeling of Switched Reluctance Machine by Coupled Field - Circuit Method

Longya Xu Eric Ruckstadter

Senior Member, IEEE Member, IEEE

The Ohio State University

2015 Neil Avenue Columbus, Ohio 43210

Wright Patterson Laboratory, AFB

Dayton, OH 45433-6563 Department of Electrical Engineering w u p o o c - 1

Abstract: Transient analysis of a switched reluctance machine (SRM) system is complicated by its unconventional non-sinusoidal operation and highly nonlinear characteristics. The analysis, however, is essential not only for optimizing the SRM magnetic structure but also for proper control of the associated power electronic circuit. In this paper, a direct modeling method for analysis of a SRM system including the power electronic converter, control and the nonlinear magnetic field of the SRM is established. The finite element method is used to model the nonlinear magnetic field, and is coupled to the circuit model of the SRM overall system. Assumptions of current density in FEA and various types of flux-current characteristics in circuit analysis are eliminated. With simultaneous computation over the entire system, the computer model provides abundant information regarding the SRM system. Experimental results are presented to prove the accuracy of the model.

Key Words: Switched Reluctance Machine, Modeling and Simulation, Coupled Field and Circuit Analysis, Finite Element Method

1. Introduction

Although induction machine drives are still the workhorse of industry, the switched reluctance machine (SRM) drive has been actively researched for more than a decade with very promising results [l-31. The SRh4 has a simple and rugged construction as well as very good overall performance over a wide torque-speed range. Recently, doubly-salient switched reluctance machines have been found to be a favorable alternative to more conventional PM synchronous and induction machines in converter fed variable- speed drives and generating systems. The fundamental feature of this type of motor system is that the SRM requires only unidirectional current and thus the converter topology and corresponding switching algorithm of the power converter is greatly simplified. In addition. unidirectional current of the SRM

system. For example, the firing and extinguishing angles of a phase current have been found to be very critical for machine efficiency and torque performance[6]. However, theoretical and experimental experiences have indicated that phase angle control of SRM current is closely related to the load level, rotor speed, rotor position, etc.; and precise control of the firing angle requires an accurate model to fully account for their complicated relations.

The complexity of SRM modeling and analysis stems from the existence of many unique facts in operating the machine. The fwst one is the unconventional non-sinusoidal excitation of the SRM. In analyzing non-sinusoidal operation, the popular rotating field theory is no longer applicable. In effect, dealing with SRM often starts with the very fundamental physics and uses the derivations from first principle. The second one is the SRM's highly nonlinear characteristics. Due to the switched operation mode of a SRM, at any instant only a small portion of core iron and winding copper is active in making a contribution to the torque production and energy conversion, and this part of the iron is always driven into deep saturation. As the consequence of local saturation, nonlinear behavior of SRM is dominant. In addition, the local over-stress of iron and copper induces mechanical vibration and deformation, which further complicates SRM analysis[7]. The third one is the interaction between the phases at current commutation in which more than one phases of mutually coupled windings are in conduction. As a result, an irregular pattern of magnetic field is produced. The last, but not the least one, is the wide torque-speed range under the electronic feedback control which constantly changes the SRM operating point.

Modeling of a complicated SRM system has been explored by many researchers over the past years, with much effort made to develop effective ways for optimal design and operation, and one of the common approaches is to use an equivalent circuit as the basic framework. To improve accuracy of the circuit model,

. .

i " e S that the power circuit configuration is immune from CUrrent shooting-through fault, which is a particularly attractive feature for meeting high reliability requirements in aerospace power applications.

The SRM, in effect, is an advanced type of stepping motor in which the current waveform of the machine must be carefully P w i W " e d to match the variation profile of the internal magnetic field so that maximum torque/ampere is extracted [1-4]. In realizing the optimal current programming, many difficult problems must be solved, which relies heavily on the proper modeling and comprehensive understanding of the entire SRM

various methods have been applied to adapt the parameters, especially the inductance, to the operating conditions, accounting for the nonlinear characteristics of the magnetic field. A step further, instead of treating the magnetic field as the product of inductance by current, the flux linkage is selected directly as the variable, and all possible flux linkages as a function of rotor positions and current levels are pre-calculated. Then, the results are stored in a look-up table, or approximated by analytical expressions [8-9]. When the dynamic analysis of the SRM is conducted, the computer program repetitively accesses the look-up table, searching for the operating point. Interpolation is necessary for those operating points not stored in the table. The circuit model with parameter adaptation works well in general, especially when only the terminal characteristics are concerned. However, the ability of this type modeling method is limited. In particular, during the modeling process the magnetic field distribution is not available at the moment of interest since only a circuit model with lumped parameters is used. In addition, mutual coupling effects between the phases are very difficulty to be included.

For the field analysis of SRM, the finite element method is commonly used with current assumed[8,10]. While the internal details of the magnetic field are available and the electroinagnetic capability of the SRM can be evaluated, the interaction of the internal magnetic field variation versus the extemal circuit can not

95 W4 067-9 EC by the IEEE E lec t r i c Machinery Committee of the IEEE Power Engineering Society for presentat ion a t the 1995 IEEE/PES Winter Meeting, January 29, t o February 2, 1995, New York, NY. Manuscript sub- mitted December 30, 1993; made avai lable for Pr in t ing November 30, 1994.

A paper recommended and approved

0885-8969/95/$04.00 0 1995 IEEE

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be modeled because the currents are artificially assumed. To remedy the situation, the latest development in modeling SRM was given by Arkadan et al in which the finite element method and state space method, in a way, was combined[ll-121, similar to the approach successfully applied to other electric machine analysis by Demerdash et. al.[ 13-14]. Satisfactory results have been achieved to account for more complicated phenomenon, including mutual coupling and fault conditions, affected by both magnetic field and power circuit. This is a significant step towards the more direct modeling of the SRM. However, the magnetic field was modeled as the product of excitation currents by inductance, and computation of a large nonlinear inductance matrix (including DC and incremental inductance, self and mutual inductance) is needed.

While the recent research work is encouraging, clearly, more advanced modeling method of SRM is still seen in demanding. In particular, the more advanced modeling method, while being expected to retain the abilities of the existing ones, should allow simultaneous investigation of external circuit and internal magnetic field accurately. In this way the internal magnetic flux distribution in space and variation in time as well as the terminal characteristics are fully described. In this paper, a modeling method for analysis of a SRM system including the power electronic circuit, control and the nonlinear magnetic field is established. Finite element method is used to model the nonlinear magnetic field of the SRM, and is coupled to the circuit model of the overall system. With simultaneous computation and convergence of the field and circuit equations for the entire system, the computer model will not only reveal the internal electromagnetic changes instantaneously but also provide sufficient information regarding the terminal dynamics of the system.

Compared to the existing modeling methods appeared in the past SRM literature[l,2,7-9,11-14], the modeling method presented in this paper differs in several aspects: i) current and flux linkages are chosen as the variables of the circuit equations, and magnetic vector potential and current density as those of the field equations respectively. The variable equivalence is made naturally between the current and current density, and between the flux linkage and vector potentials; ii) a matrix of inductance is not needed. The direct description of magnetic field in terms of flux linkages avoids splitting inductance into self and mutual components, as well as into DC and incremental components: iii) the circuit and field equations are solved simultaneously for the entire modeling process. Thus, the interaction between magnetic field and power circuit is automatically included; iv) the operating trajectory of the flux-current is automatically formed during modeling process without a table of pre-calculated flux-current characteristics: and v) the modeling results not only provide the terminal characteristics but also the detailed internal magnetic field at the instants and locations of interests. Complicated effects due to saturation and mutual coupling are easily accounted for.

An experimental testing on the SRM with the corresponding power converter circuit has been conducted to validate the proposed modeling method. The initial results from the experimental SRM system are in a very favorable agreement with those from the model prediction, verifying the accuracy of the coupled field-circuit model. The potential of this modeling method is also discussed as the future work.

2. Review of SRM Principles and Modeling

Electromechanical energy conversion in a SRM, ideally shown in Fig. 1 is accomplished by means of a time varying magnetic flux due to the rotor movement. As well known, the torque can be evaluated by the circuit equation

or, more precisely, by the equation

where Wco is the co-energy of the SRM. This equation applies both to the circuit and the field models of the SRM.

41

U

Figure 1. Configuration and Idealized Inductance and Current

Although simple, the circuit approach in the Fig 1 suffers seriously from the following deficiencies: a) saturation is not considered; b) mutual coupling between the phases is neglected; and c) no information is given with regard to the internal magnetic field at the instants and locations of interest. Therefore, the simple circuit model is of little significance for designing and operating a SRM. To obtain a more practical equivalent circuit for modeling purpose, significant modifications in describing winding inductance, or magnetic field, have been made mainly in two aspects: a) Including saturation. This is accomplished by adapting the inductance values to the excitation levels. To fully account for saturation effect, inductance is further divided into DC and incremental components. b) Including mutual coupling. This is achieved by adding mutual inductance of the involved windings. The inductance representing the SRM magnetic field is then augmented into a matrix in which the off-diagonal elements are no longer identically zero.

Another improvement made in modeling the SRM by equivalent circuit approach is to select winding flux linkage directly as the variable. Instead of describing magnetic field as the product of inductance by current, the nonlinear relationship between the flux and current are directly expressed by flux- current curves at many rotor positions. The direct description of magnetic field in terms of flux linkages avoids splitting inductance into self and mutual components, as well as into DC and incremental components. Nevertheless, computation of the flux-current curves is practical only if one phase of windings is excited at a time, with mutual coupling being neglected.

Modeling the SRM by a modified equivalent circuit becomes useful in practice. However, the capability of this approach is still limited. In particular, the description of the intemal magnetic field is totally neglected. Hence, the important local quantities such as saturation, magnetic stress, and electromagnetic forces under various transients can not be investigated. Plus, the interaction between the magnetic field and the power circuit is not properly modeled. For these reasons, a much more comprehensive and versatile model accounting for both the SRM magnetic field internally and the associated external power circuit is clearly necessary.

3. Coupled Field - Circuit Modeling Method

In order to analyze the complicated flux pattern of the SRM and its terminal characteristics simultaneously, the coupled field-circuit modeling method can be used to solve the terminal equations and the flux equations at the same time. With the end winding effects neglected, the magnetic field of the SRM is governed by the 2-D Poisson equation:

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By solving this magnetostatic equation, the intemal distribution of the magnetic field can be obtained. However, the winding currents which determine the current density, Jo, must be known for solving the equations.

The terminal voltage equations of the SRM can be easily written according to the circuit configuration of the system, assuming four-phase stator windings

where abed = (v,,vb,v,,vd)T are the terminal voltages across the stator windings; i b c d = (ia,ib,icj,j)T are the phase currents of the stator winding; &,bcd = (&,A&,b)T are the flux linkages of the stator windings; and "r" is the winding resistance of the stator. "T" indicates a transposed matrix. To find terminal current, the flux linkage variation as a function of time must be known. As can be postulated, in solving the voltage equations, the most difficult part is to determine the flux linkage variation due to the change of the currents and rotor positions.

The so-called coupled field-circuit modeling method is to combine Equations (5) and (6) together, and take the current (current density) and flux linkages (vector potential) as the system state variables. Because the system consists of two equations, two variables can be solved for uniquely. Rigorous mathematic derivation in formulation of the system equations, and the existence and uniqueness of the solution are out of scope of this paper and will be skiped in this paper. The purpose of selecting flux linkage as the circuit variable is two fold: a) to avoid a lengthy computation of the inductance matrix which may involve self and mutual components, DC and incremental components, and main and leakage components; and b) to provide an ease link between the magnetic field and electrical circuit.

By inspecting Equations (5) and (6), it is seen that the current vector, &d in (6) is related to the current density Jo in (5) with the SRM geometry given, and the flux linkage vector, &bcd, is related to the vector potential A, with the winding configuration given. In the proposed coupled modeling method, the basic Equations (5) and (6) are to be solved simultaneously by numerical iterations. A matrix of equations contained in Equation (5) are discretized into elemental form over the entire SRM cross-section as usual in finite element analysis for magnetostatic field analysis. To discretize Equations (6), one critical problem is to discretize the

time derivatives, - d% . The time derivative is replaced by the backward difference. That is,

&&cd = &bcd dt At A8

A8 where q = As indicated clearly by Equation (7), the time

derivative terms are properly discretized. Then, the global equation set containing Equations (5) and (7) can be obtained and solved simultaneously, provided that the terminal voltages h b c d and detailed geometry of the SRM are given.

Two important aspects need further explanations for discretizing Eq. (6) into Eq. (7). First, in Eq. (6), the flux linkage is a function of both the current levels and the rotor positions, and the time derivative, therefore, should require the full derivative of h with respect to both rotor position and current level. Equivalently, the flux linkage might change from one level to another not only because of rotor position but also because of current variations. The flux linkage change due to either variation should be included in the discretized equation. Indeed, in Eq. (7) both current and rotor position variations are considered as indicated by the term &bcd(8-A8. ig-Ag). Physically, it can be interpreted that the backward difference term in Eq. (7) represents

not only the induced speed voltage but also the induced transformer voltage. However, it is the induced speed voltage that makes contribution to the electromechanical energy conversion.

Second, the flux linkages in Eq. (6) contain the self and mutual flux linkage components which should be included properly in the discretized equation. Recall that in flux linkage evaluation, the magnetic field and the flux linkages are computed from the vector potential equation. In evaluating magnetic field, all possible currents are included and specified to the coil locations. Therefore, the resultant vector potential solutions automatically contain the components representing self and mutual flux linkages. The flux linkage derived from the vector potential is, therefore, a complete flux linkage. Furthemore, by computing the overall magnetic field and the consequent winding flux linkages, we can avoid unnecessary complexity to split flux linkages into various self and mutual components.

Figure 2 shows the computation flow chart used for the proposed modeling method of the SRM. Two major loops are designed in the algorithm, the inner current loop and the outer rotor position loop. Note that AV, included in the inner current loop, is the difference between the actually applied voltage and the computed voltage from Eq. (6) for the current assumed. As soon as AV falls within the predetermined error, the currents assumed converge to their true values which, in turn, determine the field vector potential of the SRM.

I I input da tz

change r o t o r

input i n i t i a l cur ren t and f l u x

FEM o f the f i e l d

t

t

Figure 2. Flow Chart of Coupled Field-Circuit Modeling Method .

In summary, the current density in Equations (5) and the currents in (6) are equivalent variables of the global system and so are the vector potential and the flux linkages in the same set of equations. In computing equations of the coupled field and current model, the field equations constantly communicate with the circuit equations to continue the computation till both the field and circuit equations are satisfied simultaneously. Attention is needed in formulating the incremental current to accelerate the numerical convergence process.

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Computation of winding flux linkages from magnetic plot provides a convenient method to link field analysis to the terminal circuit analysis. Flux linkages are significant because variations of flux linkages determines the back EMF, the kernel of electromechanical energy conversion represented by an equivalent circuit. In addition, if the flux linkages were calculated from field analysis, then an accurate circuit model would be obtained. The computation of flux linkage can be done conveniently using the results from finite element analysis. Note that if the magnetic vector potentials are known at each point over the SRM cross- section, then the flux linkage of a winding equals to the vector potential difference at the locations of the two sides of the winding. For a given SRM the flux linkages can be computed comprehensively to encompass all the possible operating points and then the data can be stored in a look-up table as shown in Fig. 5. During system simulation, the look-up table is repetitively accessed to search for the actual operating point.

4. Results of Theoretical Analysis

The computation procedures outlined in the Flow Chart are implemented and the finite element package, MAGNET, developed by INFOLYTICA CO is used, serving as the major subroutine. Several analysis has been conducted on an existing SRM, SR-90. The major specifications of SR-90 are listed in Table 1.

TABLE 1

Power rating 200 watts Base Speed 5000 rpm Torque 3.67 lb-in Voltage 160 volts Current 1.2 amps Efficiency 8 1.66% Stator OD 3.286 inch Rotor OD 1.784 inch Core Length 1.96 inch Poles number 8/6

The 2-dimensional geometry of the SR-90 machine is shown in Fig. 3 and the meshes for FEM over the across-section of the machine are generated in such a way that the meshes of the rotor are separable from those of the stator. When the movement of the rotor is simulated, all meshes of the rotor advance an incremental angle and are re-stitched to those of the stator. The separable meshes of the rotor from those of the stator greatly simplify the mesh generation process for different rotor positions.

Figure 3. Geometry of SR-90 Switched Reluctance Machine

4.1 Flux Distribution and Winding Flux Linkage

As the first step, with the assumed currents, finite element method is used to illustrate the way of obtaining flux distribution and winding flux linkages. Two typical flux distributions of the SR-90 machine are shown in Figs. 4(a) and (b), where the rotor axis is completely aligned and unaligned with the stator poles, respectively. Since the flux distribution plots display the details of the magnetic field over the entire cross-section, locating local saturation and magnetic force distribution by inspection of flux plots becomes straight forward. However, the flux distribution plot does not give information regarding the SRM terminal characteristics.

I I I 1 (a) Rotor is unaligned with the Stator Poles (b) Rotor is Aligned with the Stator Poles

Figure 4. Flux Distribution of SR-90 SRM

0.20 , I I I 1

0.16 h

t 0 p 0.12

3 is

0.08

0.04

non 0.0 0.5 I .o 1.5 2.0 2.5 3 0

Figure 5. Plot of Flux Linkages of the SR-90 SRM

Nevertheless, computing and storing a large amount of data for winding flux linkages is difficult. For example, to model multiphase excitation with winding mutual coupling, two or more phases of windings must be considered. The possible combinations of the current levels from different phases are vast. To evaluate every possible current combination under so many rotor positions, the computation time and storage memories may not be practical. Yet, the major limitation of this method is the lack of simultaneous description of the internal magnetic field and the terminal circuit. Furthermore, accuracy problems caused by interpolation of an actual operating point based on finite computed operating points are also noticeable.

4.2 Results of Coupled Field-Circuit Analysis

Direct modeling the SRM by coupled field-circuit method provides a solution to simultaneous computation on the magnetic field and system circuit. This method is especially suitable for investigating dynamic behavior of the SRM in which both the terminal characteristics and corresponding internal magnetic field are concerned. The SR-90 machine is analyzed by the coupled field-circuit method with the operating conditions suinmarized i n Table 2.

TABLE 2

rotor speed 5000 rpm transistor turn-on angle 30' DC bus voltage 160 volts transistor turn-off angle 45"

Note that modeling of the SR-90 at high speed (5000rpm) is selected because a). at high speed, SRM has to be in voltage operation mode in which the currents are variables to be predicted, and b) at high speed the back EMF of the machine is so large that the current response to the supply voltage is very sluggish (more rising and falling time in terms of rotor angle). As a result, the current overlap and the effect of mutual coupling between the phase are evident.

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Fig. 6 shows the current and imposed voltage at the terminal of the winding while the SRM experiences a series of flux changes. Note that although the input voltage maintained constant for 15', the phase current fluctuates around 1.65 amps after reaching its peak value at about 8' rotor rotation. Furthermore, the peak current does not correspond to the instant when the maximum flux linkage occurs . After the transistor is turned off, the phase current takes a sizable amount of time to decay (about 13").

As stated previously, the coupled field and circuit modeling method not only gives the terminal characteristics, but also reveals the internal magnetic field variation for the same instants. Fig. 7 shows a series of the internal flux distribution of SR-90 covering 30" rotation of the rotor in 3" steps. Note that the flux density in general increases over the entire cross-section of the SRM in time. However, the comer tips of both rotor and stator poles saturate quickly when the rotor moves towards the alignment with the stator. Yet, this local saturation reduces later when the rotor moves further towards alignment with the stator poles. Also noticeable is the SRM magnetic field which is fully charged at the beginning due to the previous phase current. Subsequently, the majority of the flux lines gradually shift to the current phase and the magnetic field is fully charged again after 15' of rotor rotation. Following the similar pattern, the magnetic field will be taken over by the next phase in approximately another 15" rotor rotation.

It is interesting to note the flux linkage-current trajectory of the phase winding as shown in Fig. 8 where the operating points for one phase current impulse is plotted while the rotor in rotating. As compared to Fig. 5 where a large number of the flux-linkage curves have to be computed in a "off-line" fashion with the neglect of mutual coupling, the operating points in Fig. 8 are calculated directly from the solution obtained by the coupled field-circuit method. The area circled out by the trajectory equals to the co- energy variation for the cycle. The average torque production of the SR-90 then can be computed from the co-energy variation of the phase winding with respect to the incremental rotor angle. The computed average torque in this case is 32.8 oz-in.

From the dynamic plots of the SRM flux distribution, other important variables, such as the tangential and radial forces can be conveniently computed, which is very important in evaluating

Figure 6. Predicted Voltage and Current Waveforms

0.08

0.07

h 0.06 v! > 0.05 b) g 0.04 cl 0.03 x

O M

0.01

I lM I "."- 0.0 0.4 0.8 1.2 1.6 2.0

Excitation Current (amps)

Figure 8. Predicted Flux Linkage-Current Trajectory

Figure 7. Flux Distribution with Voltage Source Control

45 1

force balance and mechanical stress for noise and deformation studies.

5. Experimental Testing Verification

For the same operational conditions as modeled in the last section, the SR-90 machine is tested in the laboratory. A picture has been taken for the experimental set-up as shown in Figure 9. The SR-90 is mechanically coupled to a computer controlled dynamometer as the load, and all voltages and currents are instrumented by a digital scope. During testing, the tested machine delivers 30 oz-in torque in steady state operation. Fig. 10 shows the oscilloscope traces of the phase current. Compared to the results fro the coupled field-circuit modeling method. It is clear that the current waveform predicted by the coupled field-circuit modeling method is in a very good agreement with that from the testing.

Figure 9. Experimental Test Set-Up Including SR-90 and Dynamometer

The average torque measured by the computer controlled dynamometer is 30 oz-in. Note that the torque measured by the dynamometer is the true electromagnetic torque minus some losses

Phase

24 30 36 42 48 54 60 66

Phase

Current

Current

Figure 10. Comparison of Measured and Tested Current Waveforms

due to windage and friction. Compared to the one predicted by the coupled circuit-field method, 32.8 oz-in, again the testing result is very close to the theoretical result. The accuracy of the modeling method is fully verified.

It is important to realize that by the time domain simulation of the SRM using the coupled field-circuit modeling method, the information is much more comprehensive and insightful compared to that from testing results. Furthermore, the flexibility and cost of this method is superior to that of the lab testing.

6. Conclusions and Future Work

Modeling of the SRM system is very complicated. The complicity stems from the highly nonlinear nature of the magnetic field and its interaction with the unconventional, electronic control of the SRM system. To fully account for the complexity, a simultaneous modeling of the SRM's internal magnetic field and external power electronics circuitry is necessary. The coupled field-circuit modeling technique presented and implemented in this paper provides a powerful tool for the situation. Through principle discussion, modeling development, computer implementation, and experimental validation, the following conclusions have been reached.

1). The conventional magnetostatic finite element analysis can be extended to include circuit analysis to investigate complicated transient behavior of SRM systems. Formulation of the problem is simple and conveys very understandable physics of the modeled system. The implementation of the modeling can be realized conveniently by using commercially available and technically matured finite element packages used for static field analysis. The programming time and cost for modeling is minimum, and the accuracy is high.

2) . The outputs from the combined field-circuit modeling method are comprehensive. In the field-circuit coupled approach, investigation is conducted in the time domain. Therefore. the solutions to the field equations naturally presents the results of the field variation both in space and in time, as opposed to the those obtained from conventional static field analysis. Furthermore. the modeling method enables the circuit current to interact with the field flux intensively under the constrains imposed simultaneously by the voltage and field equations. The results, therefore, contains all information regarding the effects of interaction between the extemal circuit and internal magnetic field.

3) . The coupled field-circuit modeling is versatile and flexible. In the field-circuit modeling of the SRM, the inputs to the system are the terminal voltages to the windings, rather than current, and the geometry and material specifications of the SRM. Both the input functions and the geometry specifications of the SRM can be changed easily according to the modeling purpose. By changing the input voltages, various electrical fault conditions can be simulated. In other situations, we can change the geometry or material of the SRM to model mechanical or material errors. Regardless of the origins of the fault or error (electrical or mechanical, intemal or extemal), the results from the modeling will be insightful not only towards the SRM but also to the extemal circuit.

4). By the field-circuit simultaneous computation, the accuracy of modeling of the SRM is improved. By other methods, computer modeling and time simulation for the SRM itself and the power electronic circuitry relies on a pre-computed look-up table. Interpolation is necessary and the modeling accuracy is reduced due to interpolation. In the field-circuit approach, the true operation point is obtained through iteration. The non-linearity between the current and magnetic flux is guaranteed by the finite element method.

The topics for SRM studies are comprehensive and can be studied effectively by the method developed in this paper. The following problems are particularly suitable to be solved by the coupled field-circuit modeling method: a). Flux and unbalanced forces under short circuit fault condition; b). Flux and unbalanced forces under open circuit fault condition; c). Determination of

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optimal firing angle for maximum torque production; d). Effects of eccentric rotor geometry on force production and the power circuit. A detailed research program on the topics is in progress and results will be reported in future papers.

Acknowledgment

This work was funded by the Air Force Office of Scientific Research (AFOSR) under the Visiting Summer Faculty Program. The authors are grateful to the Electrical Laboratory of the Wright Laboratory Aerospace Power Division for providing the equipment, facilities and support to complete this project on schedule.

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Eric J. Ruckstadter (M93) received the B.S.E.E. degree from Marquette University, Milwakee, Wisconsin in 1982. He joined the Naval Ship Weapon Engineering Station in Port Hueneme, CA where he worked on high power shipboard radar systems.

Since 1987 he has worked at Wright Laboratory, Dayton, Ohio, in the Aerospace Power Division. He is presently responsible for several programs to develop advanced componenets for aircraft power system applications.

Longya Xu was born in Hunan, China. He graduated from Shangtan Institute of Electrical Engineering in 1970. He received the B.E.E. from Hunan University, China, in 1982, and M.S. and Ph.. D. from the University of Wisconsin, Madison, in 1986 and 19'90 both are in Elecmcal Engineering.

From 1971-1978 he participated in 150 kvA synchronous machine design, manufacturing and testing for mobile power station in China. From 1982-1984, he worked as a researcher for linear electric machines in the Institute of Electrical Engineering, Sinica Academia of China. Since he came to the U.S., he has served as a consultant to several industry companies including Raytheon Co., US Wind Power Co., Pacific Scientific Co., and Unique Mobility Inc. for various industrial concerns. He joined the Department of Electrical Engineering at the Ohio State University in 1990, where he is presently an Assistant Professor. Dr. Xy received the 1990 First Prize Paper Award in the Industry Drive Committee, IEEEDAS. In 1991, Dr. Xu won a Research Initiation Award from National Science Foundation for his research project "A High- effciency, Low-cost Flexible Variable Speed Wind Power Generating System." His research and teaching interests include dynamic modeling and converter optimized design of electrical machines and power converters for variable speed generating and drive systems. He is a member of Electrical Machinery Committee of IEEE/PES, and a member of Industry Drive and Electric Machine Committees of IEEWAS.

453

Discusion

A. A. Arkadan (Department of Electrical and Computer Engineering, Marquette University, Milwaukee, WI): The authors should be commencied for attempting to trcat thc SRM and associated pow elcclronics as one system and on their appreciation of thc eoupled mpgnetie field-circuit approach p c r f o d by carlicr investigators (1-71. However with E@ to tbc p" papa the following should be notcd:

1- Some of thetcnninology used in the paperis soomehow misleading such U tkuscofthe words "Diroct" and "Coupling" in the title and the body of thepapa. This h.ppars to betbc case since tbc coupling lppmaeh used here is an 'indircd" one and is simiIarto tbc rppForbes used in di worlrs [I -q except for the constant speed. For htu details on fuu-flcdgcd dirca coupling .pprolEbcs. sec rcf.aulcu [&IO].

z 'Ihc authors used thc tcrms incrcmclrtal components, de. self. Md WItlUl todcsaii tbc indwtanccs uscd in pnviw Works [I-q.This is iocoma Ebcetbgeplpcn uscdthctcrms "incruneotal" and "appamt" todcsaibethctypeof hkances nmcrtho dtsaibingtbeirwmponu~ts. As forthc inductance rcpnsscnratio~arrialaia cq" mn used which have harmonic componcn& with mm m u avcraga for "z of& i n ~ p a r a m c t c r s .

two qluuions, two variables CM be solvd for uniquely". I l l i s rtltcmot is vay misleading

rimplisatioa iolromusd baebytbcllacof

3- wheo dwxibingquations (5) and (a) the authors state that &cpuse tkryrtemumsistsof

sincc (5) and (6) are two systcms of quations which have thc wdcl mv's (uaully numba io thc thousands) and the m n t s as unknowns.

4- The authom claim that their simulations result in tbc field (flux) vdues at MY instant of tim. Tbis cwld not be the case sincc the authors are using a sa of static fAd solutions and the rotor spcad is assumed constant Hence. these solutions m at best "snap shots' of tbc magntiC field However this method cwld be impmvcd if thc uuhar include tbc meh.nicll (spocd) quation in thcfonnulations and if the specd is msdc an in- pmtofthc aitaiafor convergence as was &ne in earlier work [5.6]. In addition, it should be noted that b e field (flux) plots wcrc always available in earlier works on such coupled field-=ircuit rdutim as can be s e n from the literature [ 1-71. Fu~thermore. it should be cmphasizsa that since the authors appear to have assumed UHLstant speed, their simulation can not rerult in rulirtic inst~tuuous values of the toque or internal field conditions. Accordingly. the intcnctiOn buwcco tbc e x t a d switching c h i t s and the intend magnctic fwlds CM uot be properly modelad fa this class of machines.

5- The authors claim that their approach is more efficient (computationally) compared to thc other approaches. However. based on the dewiflion given in the papa. it seems that the coupling (interfacing) behv,an the circuit quations (6) and the field quations (5) is doae manually. That is, quation (6) is solved first and the results a~ then uscd to solve equation (7). This is contrary to the approach presented in [5.6] whcn the field and circuit quuiol, dvar pn integrated in one program so that thhcn is no nud to manually pass mults from one algorithm to the othcr. Acoordingly. it seems that the work presented by the authors docl wt accomplish any savings in terms of the computation time in comparison to carlicr work^. For details. PI should be consulted for information on computation timcs.

6- Finally. as the SRM reporled on in this work is thc SR-90 which usually employs cumnt chopping in its controller. the autkm arc kindly r q w t e d to shed som light on thc "I used to modcl the current chopping in their analysis. An inspection of the "d ament waveform of Fig. 10 m v d s some effeas of chopping or noise. Homva this effect is not cku from thc calculated cumnt waveform given in thc same figure.

ReIvcnas

of s.tuntcd V d w of [I1 T.W. Nchl. F.A. FOWL and N.A. Dcmerdssh. N.A.. "- I E E E T m . PmverAppr. rmdSyr..Vol. lOl.pp.6141451.1982. [21 AA.ArLadsn.T~~jaz iandN.A.~"Computcr-AidedModcl ingofaRkt i f*d DCLoad P a " t MagnUGencntor Systcm with MultipleDampa WindiDgs in mC N*ml I b c ~ o f r ( C f a m a , ' I E & E T m . E n e r . cOnv..Vol.4.No.3. pp.518-525.segL 1989. C31 RWang. R, and NAD. Dcmrdash, "Canputatioo ofW Pafomrwe d OQr paramdas of Extra High Speed Modified Lundcll Altcmrtors from 3DFE Magodic Rcld

[41 AA. ArLadan and B.W. Kielgas"Effcc~~ of FocccFittingon thc Induasna Rofi*c€a Switfbed Rcluctaaa Motor." IEEE T m . Magn. Vol. 29. No. 2 pp. zMMzoo9. Mar. 1993. [q A. A. Arhdsq mdB.W. Kidgas.SwitcbcdRrhcnnaMacorDrivc Syrolnr DynrmicRdorrmDDc Redic(im md J3wi"taI V-m" V d 9. No. 1. pp. 36-44. Mr. 1994. 161 AA. ArLadsn. and B.W. Kielgas. "Switched Rcluctmce Motor Drive System Dynamic ~ ~ P F t d i a i o n U n d e r I n t c m a l a n d ~ a l F P u l t C o n d i t i o a g ' I E E E T ~ . E n c r . Canr. Vol. 9, No. 1, pp. 45-52. March 1994.

AA. Arl-sdan, and B.W. Kielgas. "The CMlpled Robkm in Switcbed Reluarnee Mota Drive Systems During Fault Conditions."[EEE Trans. Maen, Vol. 30. No. 5, pp. 3256-3259. scpt 1994. [SI E.G. Strangas, 'Coupling thc Cimit Equations to the Non-linear T i m Dcpendmt M d Solution in InvCaaDriven Induction Motors," IEEE Trans. Maan.. Vol. 21, No. 6. pp. 2408- 2411,1985 191 F. Piriou and A. R a d & "Coupling of Sahuated Elcctmmagnetic Systems to N a ~ U n m r Power Eleaxonic Devices." IEEE Tram. Magn.. Vol. 24. No. 1. pp. 274-277, Jan. 1988. [IO] A.A. Arlradan and RV. VanderHciden, "Thne Dimensional Nonlinear Finite EIcmnt Modeling of a Voltage Soura Excited Transformer Feeding a R d f w r Load." IEEE 7ian.r. Magn., Vol. 28. No. 5, pp. 2265-2267, Sept. 1992.

. . Rotating M.chineryIncrc"tal a n d e t Inductanm by anEnaey Pat" -v

SOlUrioa$" (\gn IBEB TraaS Ena. Coav.), Vol. EC-7. NO& pp. 342-352.1992

Manuscript received March I, 1995.

L. Xu and E. Ruckstadter:

The authors wish to thank Dr. Arkadan for his interests in the paper, and would like to take the opportunity to clarify the issues raised in his discussion. The closure follows the sequence of the discussion.

1) "Direct modeling" used in this paper is for purpose of comparison to the methods used for SRM in the literature, including [1,2] mentioned by the discusser. The fundamental difference between the methods in the paper and in [ 1.21 is how to model and how to interface the SRM magnetic field analysis to the circuit analysis. In [ 1.21 the flux linkage is treated as a product of current by incremental inductance. The incremental (self and mutual) inductance is pre-calculated at no-load and rated-load levels in an off-line fashion by FEA and decomposed into Fourier series caithout too much physical meaning). However, in SRM reality, when the two phase currents are simultaneously in conduction and interact with each other, magnetic mutual coupling between two phase occurs. Clearly, the mutual coupling interaction occurs at many possible current level combinations. It is seen that mutual coupling between the phase windings is improperly considered in [1,2] since the current levels selected for inductance computation are far removed from SRM reality. In addition, the saturation effect is based on inductance computed at no-load and rated-load levels, which again is remote from the real situation. It is from these observations, our paper directly selects flux linkage as the circuit variable and directly couples the winding flux linkages to the mvp computation of the field in each time step. Compared to the method presented by the discusser, it is believed that the method in our paper is more direct and closer to the reality for the mutual coupling between the phases and the saturation effect. In effect, the principles of direct finite element analysis and modeling methods have been introduced for more than a decade[3] and have been accepted ever since. This is evidenced by many successful applications of the method in evaluating several special electric machines 1451.

2) With regard to the terms used for the description of SRM inductance, it should be noted that these terms are actually referred in pairs in our paper as: DC and incremental inductance, self and mutual inductance, main and leakage inductance, etc. In the paper no cross refemng is implied among these pairs. In the first pair, the DC inductance is sometimes called "apparent inductance", or "effective inductance". These terms have been used by IEEE professionals for many years. On contrary, Fourier decomposition of incremental inductance into DC and harmonics components as a function of rotor positions as done in the discusser's paper is rarely seen and lacks physical meaning. This is, perhaps, where the discusser's confusion regarding the termonology is connceived.

3) As we stated in the paper, because the system consists of two equations, Eqs. (5) and (6), two variables can be solved uniquely. Obviously, this statement is complete and correct. However, one can not simply interpret this statement into a trivial case disregarding the finite element method and matrix form of electric machine equations. In both equations, each variable is a vector representing many entries. It appears to the authors that the statement in the paper did not confuse the discusser at all, and we can only speculatewhy he chose to criticize.

4) As concluded in the paper, one of the major features of the coupled field-circuit modeling method of SRM is that the results not only provide the terminal characteristics but also the detailed intemal magnetic field at the moments and locations of interests. The discusser contends that "this can not be the case since the authors are using a set of static field solution and the rotor speed is assumed constant. ... these solutions are at best 'snap shots' 'I. The discusser is correct in that the results are "snap shots". Nevertheless, he perhaps did not realize that the "snap shots" are SO frequent that it is only limited by the mesh size in the airgap, not by the method itself or by an assumption of constant rotor speed. The discusser is also correct in that if the mechanical equation is included, the speed dynamics can be simulated. However, in order to highlight the interaction between circuit and field analysis, it is the authors intention to choose constant speed conditions.

454

Practical operation of a SRM can justify such a constant speed operation mode. Note that with a constant speed, the SRM torque could have high ripples. In such a condition, while the ripple components of the torque am absorbed by the rotor inertia, the average torque is balanced by the load. Given the magnetic flux distribution at the instant of interests, we can always conveniently evaluate SRM torque by many methods [6,7]. The same principle applies to variable speed operation mode of a SRM.

5) It appears to the authors that the discusser has not studied the details of the coupled field-circuit method illustrated by the flow chart in Figure 2 before his assumption that the coupling (interfacing) between field and circuit is done manually. It is equally unclear how the discusser defined his "manual" or "automatical" interface between field to circiut.

6) With respect to the final comment made in the discussion, the discusser is referred to [8-101 to examine why, when and how

choD -Ding (not current chopping) is applied to regulate current in SRM. When the SR-90 is operated at 5000 rpm with rated torque, the converter bus voltage can not be chopped due to the high back EMF. The tiny ripple of the recorded current is not due to any chopping and is very normal from any high accuracy, modem digital scope measurement. The discusser is again referred to the paper, particularly, the paragraph in page 4 right behind Table 2 for the non-chopped voltage operation mode.

References:

1). A. Arkadan and B. W. Kielagas, "Switched Reluctance Motor Drive Systems Dynamic Performance Prediction and Experimental Verification," IEEE Transactions on Energy Conversion, Vol. 9, No. 1 March 1994, pp 36-44

2). A. A. Arkadan and B. W. Kielagas, "Switched Reluctance Motor Drive Systems Dynamic Performance Prediction Under Intemal and External Fault Conditions," IEEE Transactions on Energy Conversion, Vol. 9, No. 1 March 1994, pp 45-51

3). T. Nakata and N. Takahashi, "Direct Finite Element Analysis of Flux and Current Distributions under Specified Conditions," IEEE Transactions on Magnetics, Vol. 18, No. 2, March 1982,

4). S. Nonaka, K. Kesamaru and K. Horita, "Analysis of Brushless Four-Pole Three Phase Synchronous Generator Without Exciter by the Finite Element Analysis." IEEE Transaction on Industry Applications, Vol. 30, MaylJune 1994, pp 615-620

5). S. Nonaka and K. Kesamaru, "Analysis of New Brushless Self-Excited Single Phase Synchronous Generator by the Finite Element Analysis," IEEE-IAS Annual Meeting, Coference Proceedings, Oct. 4-9, Houston, 1992, pp 198-205

6). P. Hammond, "Energy Methods of Electromagnetism," Clarenson Press. Oxford, 1981

7). K. J. Binns, C. P. Riley and M. Wang, "The Efficient Evaluation of Torque and Field Gradient In Permanent Magnet Machines with Small Airgap." IEEE Transactions on Magnetics, Vol. 21, No. 6, Nov. 1985, pp. 2435-2438

8) T. J. Miller, "Switched Reluctance Motors and their Contro1,"Magna Physics Publishing and Oxford Univ. Press. 1993

9). T. J. Miller and T. M. Jahns, "A Current-Controller Switched Reluctance Drive for FHP Applications," Proceedings of Conference on Applied Motion Control, Minneapolis, June

10). L. Xu and T. A. Lipo. "Analysis of a Variable Speed Singly- salient Reluctance Motor Utilizing Only Two Transistor Switches", IEEE Transactions on Industry Applications, Febwarch, 1990, pp. 229-236

pp 325-330

1986, pp. 109-117

Manuscript received April 19, 1995.