simulation Paper 5

92 IEEE TRANSACTIONS ON EDUCATION, VOL. 41, NO. 2, MAY 1998

Re-Inventing the Electrical Machines CurriculumMalcolm W. Daniels,Member, IEEE, and Randall A. Shaffer,Member, IEEE

Abstract—Undergraduate courses in electromechanical energyconversion are typically oriented toward the steady-state analysisof electrical machines. The advent of low-cost computer powerand the availability of numerical software tools provide the op-portunity to fundamentally reorient the pedagogical approach tothe topic. A new approach is consistent with the developing needto emphasize the study of new machine designs and machinesemployed for control purposes as well as machines for use in moretraditional power applications. This paper presents the resultsobtained from simulations developed as an integral part of anundergraduate electrical machines course at the University ofDayton. Sample simulation files are presented to demonstrate theease with which the matrix model of the machine is transferred tothe program. The matrix models and simulation results of the fol-lowing machines are included: the single-phase transformer, thevariable-reluctance machine, the cylindrical-rotor dual-windingmachine, the symmetrical two-phase induction motor, a PWM-controlled dc machine, and an inverter-driven brushless machine.The selected machines provide a systematic framework for analy-sis and simulation and present problems of increasing complexityto the student.

Index Terms—Electrical machines, motors, simulation.

I. INTRODUCTION

UNDERGRADUATE courses in electromechanical energyconversion are often regarded by students to be historical

artifacts of electrical engineering. Indeed, it has so long beenconsidered a classical topic that many industries have noted adistinct lack of new engineers entering the field. Those who areinterested in the topic seem to be a dying breed. The motivationto re-invent the electrical machines curriculum is largelyattributed to the desire to modernize the student’s perceptionof the subject. The accessibility of low-cost computing powerand the availability of sophisticated numerical software toolsprovide the opportunity to reorient the pedagogical approachto the topic.

Several factors combine to motivate a redefinition of theundergraduate course: Changes in the student profile, in-ternal competition, laboratory modernization, and changingemployment prospects. The engineering profession has longbeen a pioneer in regard to the integration of computers inthe curriculum as well as in teaching. However, computingresources have yet to be significantly incorporated into thesubject of electrical machines, as is evidenced by the nature ofthe textbooks that support this type of course. A brief reviewof current texts from the major publishers reveals that, to a

Manuscript received August 7, 1995; revised December 29, 1997.M. W. Daniels is with the Electrical Engineering Department, University

of Dayton, Dayton, OH 45469-0226 USA.R. A. Shaffer is with the Electrical Engineering and Computer Science

Department, Embry-Riddle Aeronautical University, Prescott, AZ 86301 USA.Publisher Item Identifier S 0018-9359(98)03511-0.

large extent, the computer is rarely considered a support toolfor the study of the subject. Furthermore, many of the availabletexts propose a “device-oriented” approach to the topic ratherthan one based on a theoretical development. Indeed, it isdifficult to find an explanation or even a brief discussion of ageneral theoretical framework in which to describe the subject.There are a few notable exceptions: Krause and Wasynczuk[1], Krause, Wasynczuk, and Sudhoff [2], and the much usedand revised text by Fitzgerald, Kingsly, and Umans [3] retaina somewhat generalized approach to the topic. An older textby Thaler and Wilkox [4] is also noteworthy.

Today’s students expect to use the computer as a toolto both understand the conceptual issues in any subject aswell as to facilitate analysis and design. In the prospect ofattracting students, a modernization of both the course and thesupporting laboratory is essential. A recent paper by Nehrir,Fatehi, and Gerez [5] observes that the wrong impression madeon students in the first power course deters them from takingfurther electives in the subject. Nehriret al. maintain thatcomputer simulation complements the laboratory experienceand stimulates interest and participation in the lecture course.Additionally, students are now increasingly likely to encounterelectrical machines in nonclassical environments. Even thoughthe majority of students will work in other areas, many ofthem will use machines of various types. The traditionalsinusoidal and direct-current steady-state solutions will, inmany instances, be inappropriate for their tasks. Indeed, theutility of the development of such solutions is, perhaps, inquestion. Demerdash, Luo, Alhamadi, and Mattingly [6] makethis same observation in their recent paper in which theyaddress the same topic.

Developments on the integration of computing resourcesare reflected in courses implemented at several institutions.Demerdashet al. describe the electrical machines course atClarkson University. They argue the benefits of academiccomputing in the teaching of electric machinery and theirassociated electromagnetic fields to undergraduate senior andgraduate students. Their program is adventurous as it includesan integrated use of finite-element machine analysis and time-domain simulations of various machines. Riaz [7] discussesthe use of MATLAB to simulate induction and synchronousmachines and provides steady-state and transient solutionsto the transformed machine models. Alvarado, Canizares,Keyhani, and Coates [8] discuss the issue of various declar-ative languages for the study of electric machines. Theirdiscussion includes a comparison of the use of PC-MATLABto other languages. Belmans and Geysen [9] discuss course andlaboratory developments in electrical machine and machinedrive design at the Katholieke University, Leuven. Chan [10]

0018–9359/98$10.00 1998 IEEE

DANIELS AND SHAFFER: RE-INVENTING THE ELECTRICAL MACHINES CURRICULUM 93

also reports the use of spreadsheet packages to automate ma-chine calculations, although the technique is primarily orientedtoward steady-state solutions. The integration of computers toelectric machines laboratories has been reported by severaluniversities: Krein and Sauer [11] at the University of Illinoisat Urbana-Champaign, Mohammed and Gordon [12] at FloridaInternational University, and Kasten, Kent, Mako, and Turnerat Ohio State University [13] all describe recent upgradesand renovations of their electric machines and power systemslaboratories.

In an effort to address the need to modernize the machinescurriculum, this paper presents the numerical solutions tomachine models used to support an undergraduate electro-mechanical machines course in the School of Engineering atthe University of Dayton. The objective is to provide a sys-tematic method for simulation of the transient and steady-stateoperation of different machines. These solutions are developeddirectly from the generalized machine matrix models. Refer-ence frame transformations on the models are circumvented,which reduces the complexity of the theoretical presentationand facilitates the study of the topic at the undergraduate level.The course parallels the approach developed in [1].

The following machines are considered: The single-phasetransformer, the elementary reluctance machine, the dual-winding machine, the symmetrical two-phase induction ma-chine, the shunt-wound dc machine, and the brushless ma-chine. The simulations illustrate the reluctance and alignmenttorque mechanisms that result from spatially varying airgapsand distributed windings. Practical simulations illustrate thetransient and steady-state response and operation of variousmachines controlled by modern techniques. Machine operationfrom pulsewidth-modulated (PWM) dc supplies, frequency-controlled inverters, and other common circuit topologies arereadily addressed.

The solutions to the generalized machine equations are ob-tained from the Runge–Kutta integration algorithms in MAT-LAB [14]. The selection of MATLAB as an appropriate tool isjustified by student familiarity, ease of use, accessibility, andavailability of a student version of the software, retention ofa declarative programming environment, and the clarity andsimilarity of the software solution to the theoretical problem.The theoretical simplicity of the generalized machine approachis maintained by a systematic numerical solution of couplednonlinear differential equations. The solution algorithm avoidsmatrix inversion and other numerical difficulties. The simu-lations include plots of the rotor and stator currents, statorvoltages, rotor position and velocity, and the torque–speedcharacteristic of the machine.

II. THEORETICAL FRAMEWORK

With familiar assumptions in regards to the magnetic lin-earity of materials used in machine construction, a state-spacemodel for each machine is developed in the form of a series

– circuit

(1)

in which is the winding supply voltage vector,is the wind-ing current vector, and are square matriceswhose elements represent resistive and inductive parameters,respectively, and is the generated back electromotive force(emf). With the exceptions of the transformer and the brushlessmachine, the elements of the resistance and inductance matri-ces generally vary with time. Also, the back emf is zero for allmachines considered except the brushless machine. In additionto the electromagnetic equations, a second-order differentialequation is assumed to model the mechanical characteristicsof the machine

(2)

in which and are the effective inertia and dampingcoefficients, respectively, and is the load torque.

The solution of the machine equations is obtained after anelementary reorganization of the model equations. With thederivative of the current vector expressed in terms of thewinding voltage sources and the lumped model parameters,the state-space model becomes

(3)

The mechanical equation of the machine is expressed instate-variable form as

(4)

(5)

The current vector of the machine and the state variablesand comprise the entire state vector which must be

solved for by the integration routine. When practical,is determined algebraically to reduce the simulation time andto avoid ill-conditioned matrices. In addition to the solutionof the state vector, the electromagnetic torque of the machinemust also be computed as the machine equations are solved.

The book by Krause and Wasynczuk provides a particularlylucid introduction to the matrix modeling approach to machineanalysis and is perhaps the only introductory machines textwhich does so. Accordingly, the following summary machinedescriptions are deduced from the generalized models devel-oped in [1].

A. Single-Phase Transformer

(6)

(7)

(8)


B. Variable-Reluctance Machine

(9)

(10)

(11)

C. Dual-Winding Machine

(12)

(13)

(14)

is formulated the same as (8). for the cylindricalrotor machine.

D. Symmetrical Two-Phase Induction Machine

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

E. Shunt-Wound DC Machine

(25)

(26)

(27)

(28)

F. Brushless Machine

(29)

(30)

(31)

(32)

(33)

(34)

(35)

In the generalized machine models, the spatial variations ofthe inductive parameters are assumed negligible. The inclu-sion of more complex inductance profiles would require, forexample, a finite-element approach to the modeling problem.

III. D ISCUSSION OFCOMPUTED RESULTS

The first program developed by the students is a transientsimulation of a single-phase transformer. The transformer isfamiliar to the students from previous classes in circuit theoryand from practical exposure to the device in the laboratory. Thedevelopment of the matrix model of the transformer provides aconvenient vehicle to introduce matrix models of machines aswell as the mechanics of their solution using MATLAB. Theexample also provides a gentle introduction to the simulationproblem. A range of test results can be developed from thesolution of only one or two differential equations. Typical


(a) (b)

(c) (d)

Fig. 1. Transient simulation of the variable-reluctance machine.Bm = 0:1, LA = 0:1, LB = 0:1, Lls = 0:01, J = 0:001, Rs = 10, V = 120,TL = 0. Initial rotor position 45� (solid line) and 350� (dashed line). (a) Winding current. (b) Rotor angular velocity. (c) Torque–speed characteristic.(d) Rotor angular position.

results obtained from simulations include the standard open-circuit and short-circuit tests, load tests, and instantaneousshort-circuit tests under loaded conditions. The steady-statesolutions are developed from an analysis of the equivalentcircuit of the transformer to verify the numerical solution ofthe equations.

One of the major advantages of the use of dynamic simula-tions in the introductory course is the ability to illustrate funda-mental concepts associated with machine operation. A studyof the elementary reluctance machine and the dual-windingmachine illustrates the two major torque mechanisms onwhich most machines depend, specifically, reluctance torqueand alignment torque. The elementary reluctance machine,comprising a single winding on the stator and a salient rotor, issimulated with different initial rotor positions. The simulationresults for initial rotor positions of 45and 350 are shownin Fig. 1. The results illustrate the development of torquewhen the rotor is unaligned with the minimum-reluctancepositions. The action of the torque is to align the rotor and,consequently, minimize the stored energy in the magnetic fieldof the machine at the equilibrium positions 90and 270.The cylindrical rotor dual-winding machine consists of two

interacting windings, one on the stator and the other on therotor. The dual winding machine simulations illustrate thedevelopment of torque when the principle magnetic axes ofthe rotor and stator windings are unaligned. The action ofthe developed torque is to rotate the rotor of the machine toproduce alignment. Other results can be readily obtained fromthe simulations such as the use of ac and dc sources and theaddition of rotor saliency. For the dual-winding machine, thestudent solves four coupled nonlinear differential equationsvia MATLAB with only a minor increase in programmingcomplexity from the transformer simulation program.

One principle advantage of the use of MATLAB is that itpermits a nearly direct translation of the machine models fromtheir mathematical expressions to the simulation program. Thefacility of the translation is illustrated in the simulation filesfor the dual-winding machine given in the Appendix. In thefunction file, the elements of matrices (12) and (13) are readilyvisible, as are (3), (4), (8), and (14). The use of the MATLABode45 function (not shown) makes the method of integrationvirtually transparent.

The first practical machine investigated by the student isthe two-phase symmetrical induction motor. The advantages


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 2. Transient simulation of two-phase symmetrical induction machine.Bm = 0, J = 0:026, Lr = 0:0361, Ls = 0:0361, Lm = 0:0352,P = 2, Rr = 0:201, Rs = 0:295, TL = 0 (initially), TL = 5 for t > 0:6: (a), (e) Stator current, phaseA. (b), (f) Rotor current, phaseA.(c), (g) Rotor speed. (d), (h) Torque–speed characteristic.

of the use of a matrix formulation of the equations becomeapparent with the induction motor because the machine modeland the associated mechanical load are described by a set ofsix equations. The simulation is accomplished directly fromthe generalized machine model without the use of variabletransformations to facilitate the solution. In contrast, Alvaradoet al. [8], in their comparison of different languages for use

in machine simulations, use the transformed model in theirevaluation. As with the transformer simulation, a range ofresults is readily developed with minor program changes suchas the effects of source voltage imbalance and load torquevariations. A transient simulation of the machine started fromstall is shown in Fig. 2(a)–(d). The simulations reveal manyof the pertinent characteristics of induction machine operation


(a) (b)

(c) (d)

Fig. 3. Bm = 6:04E-6, J = 1, LAA = 0:012, LAF = 1:8, LFF = 120, Ra = 0:6, Rf = 240, Va = 240, 50% duty cycle,Vf = 240. (a)Armature voltage. (b) Armature current. (c) Rotor speed. (d) Torque–speed characteristic.

without the pursuit of a steady-state equivalent circuit. Thesimulation is continued in Fig. 2(e)–(h) in which a 5 Nmstep-change in load torque occurs at 0.6 s. The resultsillustrate the dependence of the steady-state operating speedon the load conditions. Also illustrated is the concept of slip,shown by the difference in the frequencies of the rotor andstator currents.

The potential for a departure from the traditional machinescurriculum is possible with minor simulation program changes.Many of the operating characteristics of the newer machinescan be deduced from appropriate simulations. It is also possibleto preview more complicated operating conditions, particularlythose associated with power electronically controlled supplies.Once the students investigate the operating characteristics ofthe shunt-connected dc machine supplied by a dc source,the operation of the motor from a PWM-controlled supplyis considered next. The transient response of the dc machineunder PWM-control is shown in Fig. 3. The results illustratethe relationship between duty cycle and average rotationalspeed of the motor. Also revealed in the results is the high-frequency ripple content in the armature current, and conse-quently in the electromagnetic torque, which illustrates one ofthe disadvantages of modern machine control techniques.

Finally, the brushless machine operated from a variable-frequency supply is investigated. After an investigation of thecharacteristics of the machine under sinusoidal excitation, thestudents simulate the inverter-driven brushless motor. Resultsof the inverter-supplied machine simulation are shown inFig. 4. The results illustrate how the brushless machine, whichis actually a permanent-magnet synchronous motor, behaveslike a dc shunt motor in that the operating speed is directlyproportional to the source voltage.

IV. CONCLUSION

The availability of modern computing resources makes theupdate of the machines curriculum not only possible, but alsonecessary, in order for schools to remain competitive andmeet the expectations of the modern student. The subject ofelectrical machinery can no longer be limited to the study ofequivalent circuits and steady-state operating characteristics.

The use of MATLAB and the method whereby the matrixmodels of the machines are directly translated to the simulationenvironment have been very successful in the course and havebeen well received by the students. The systematic methodprovides a mechanism for machine simulation directly fromthe generalized models of the machine, while the essential


(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 4. Bm = 0, J = 0:0001, �m = 0:0826, Lss = 0:0121, Lms = 0:011, P = 4, Rs = 3:4: (a), (e) Stator voltage, phaseA. (b), (f) Statorcurrent phaseA. (c), (g) Rotor speed. (d), (h) Torque–speed characteristic.

structure of each model is retained in the associated simulationprogram.

The simulation exercises also demonstrate the tradeoff be-tween numerical accuracy and computation time. In somecases, particularly where the machine is supplied by a high-frequency switched source, the simulation times can be fairlylong. In these instances, the students are encouraged to in-crease the error tolerance, especially during the debuggingstage of programming. The students are also encouraged to

save the simulation data for certain types of postprocessing,such as cubic spline interpolation and the use of customplotting routines.

A logical extension to the course would be to includemore complex expressions for the inductive parameters inthe machine models. A finite-element toolbox, in conjunctionwith MATLAB, would be appropriate to extend the conceptualissues covered in the course and allow spatial variations to beconsidered in the model parameters.


APPENDIX

Main Program (Filename: dwmac.m; typedwmacat the MATLAB prompt)

close all, clear all, clcglobal Bm J LA LB Lls Lr Lsr P Rr Rs

% Machine parameters

Bm = 0.05;J = 0.001;LA = 0.1;LB = 0;Lls = 0.01;Lr = 1;Lsr = 0.1;P = 2;Rr = 10;Rs = 10;

% Simulation

X0 = [0; 0; 0; pi/4];tspan = [0 0.4];[t, X] = ode45(‘dualwind’, tspan, X0);

% Extract machine variables and save in file called dwmdata

Is = X(:, 1); Ir = X(:, 2); wr = X(:, 3); thr = X(:, 4);Te = Lsr Ir Is cos(thr);save dwmdata t Ir Is Te thr wr

% Plot the results

subplot (2, 2, 1), plot (t, Ir, t, Is), title (‘Winding Currents’)xlabel (‘t (sec)’), ylabel (‘(A)’), gridsubplot (2, 2, 2), plot (t, wr), title (‘Rotor Velocity’)xlabel (‘t (sec)’), ylabel (‘(rad/s)’), gridsubplot (2, 2, 3), plot (t, thr), title (‘Rotor Position’)xlabel (‘t (sec)’), ylabel (‘(rad)’), gridsubplot (2, 2, 4), plot (wr, Te), title (‘Torque-Speed Characteristic’)xlabel (‘wr (rad/s)’), ylabel (‘(Nm)’), grid

Function File (Filename: dualwind.m)

function dX = dualwind (t, X)global Bm J LA LB Lls Lr Lsr P Rr RsIs = X(1); Ir = X(2); wr = X(3); thr = X(4); I = [Is; Ir];

% Machine equations

Vr = 120;Vs = 120;V = [Vs; Vr];TL = 0;R11 = Rs + 2 wr LB sin(2 thr);R12 = wr Lsr cos(thr);R = [R11 R12; R12 Rr];L11 = Lls + LA LB cos(2 thr);L12 = Lsr sin(thr);D = L11 Lr L12 L12;Li = 1/D [Lr L12; L12 L11];dI = Li [V R I];Te = Lsr Ir Is cos(thr);dwr = P/(2 J) (Te TL) (Bm/J) wr;dX = [dI; dwr; wr];


REFERENCES

[1] P. C. Krause and O. Wasynczuk,Electromechanical Motion Devices.New York: McGraw-Hill, 1986.

[2] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff,Analysis of ElectricMachinery. New York: IEEE Press, 1995.

[3] A. E. Fitzgerald, C. Kingsley, and S. D. Umans,Electric Machinery.New York: McGraw-Hill, 1990.

[4] G. J. Thaler and M. L. Wilcox,Electric Machines: Dynamics and SteadyState. New York: Wiley, 1966.

[5] M. H. Nehrir, F. Fatehi, and V. Gerez, “Computer modeling forenhancing instruction of electric machinery,”IEEE Trans. Educ., vol.38, pp. 166–170, May 1995.

[6] N. A. Demerdash, Z. Luo, M. A. Alhamadi, and B. T. Mattingly, “Teach-ing electrical machinery and associated electromagnetic fields—A casefor the benefits of academic computing,”IEEE Trans. Educ., vol. 36,May 1993.

[7] Riaz, “Computer-aided teaching of electric machines using MATLAB,”in Proc. Int. Aegean Conf. on Electric Machines, May 1992.

[8] Alvarado, Canizares, Keyhani, and Coates, “Instructional use of declara-tive languages for the study of machine transients,”IEEE Trans. PowerSyst., vol. 6, no. 1, 1991.

[9] R. Belmans and W. Geysen, “Impact of the computer developmentson the education of engineering graduates in electrical machines anddrives,” presented at MELECON‘89, 1989.

[10] T. F. Chan, “Analysis of electrical machines using symphony,”IEEETrans. Educ., vol. 35, Feb. 1992.

[11] Krein and Sauer, “An integrated laboratory for electric machines, powersystems, and power electronics,”IEEE Trans. Power Syst., vol. 7, no.3, 1992.

[12] Mohammed and Gordon, “Analysis of rotating machine concepts inthe energy conversion laboratory from experimental data,”IEEE Trans.Power Syst., vol. 6, no. 2, 1991.

[13] Kasten, Kent, Mako, and Turner, “Modernization of machines labora-tory,” in Proc. Amer. Power Conf., 1992, vol. 54, p. 1.

[14] MATLAB Manual, The Mathworks, 1994.

Malcolm W. Daniels (M’86–M’91) received the B.Sc. and Ph.D. degreesin electrical and electronic engineering from the University of Strathclyde,Scotland, U.K., in 1979 and 1983, respectively.

After working in the industrial control area, he joined the Industrial ControlCenter at the University of Strathclyde, where he worked on Backup RollEccentricity Control problem in the steel industry. He joined the Universityof Dayton, Dayton, OH, in 1986 and has since been involved in researchand teaching in the Department of Electrical and Computer Engineering. Hiscurrent interests are industrial control problems related to electrical machines,VSC control systems, and educational philosophy.

Randall A. Shaffer (A’90–M’91–S’94–M’96) received the B.Sc. degreein electrical and electronic engineering from California State Universityat Sacramento in 1984, the Master of Engineering degree from CaliforniaPolytechnic State University at San Luis Obispo in 1989, and the Ph.D. degreein electrical engineering from the University of Dayton, Dayton, OH, in 1996.

He is currently an Assistant Professor at Embry-Riddle AeronauticalUniversity, Prescott, AZ. His interests are in power electronics, motor andcharge control, and variable-structure systems as well as sliding-mode control.

Dr. Shaffer is a member of IEEE Societies on Automatic Control, ControlSystems Technology, Power Electronics, and Education.

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