IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 43, NO. 5, MAY 2015 1381

Modeling and Analysis of HomopolarMotors and Generators

Thomas G. Engel, Senior Member, IEEE, and Evan A. Kontras, Graduate Student Member, IEEE

Abstract— The dc homopolar motor converts electrical energyinto mechanical rotational energy using the Lorentz force. Thesame machine can be operated in reverse to convert mechanicalenergy into dc electrical energy. To better understand homopolarmotors and their suitability for use in various applications, acomputer model was created using PSpice. Forces opposing themotor rotation include back voltage, eddy currents, moment ofinertia, and sliding contact friction and are analyzed in detail.Forces and torques are discussed and calculated analytically. Thecapabilities of the homopolar machine operating as both a motorand a generator are considered. Using current research fromthe University of Missouri on helical guns and railguns, whichutilize similar electromagnetic forces for linear acceleration, themaximum efficiency of the homopolar motor during transientstart-up phase is examined. The measured homopolar motorefficiency in this paper asymptotically approaches 50% and isdetermined by several variables. Experimental data are collectedand used to compare the simulation results and verify theaccuracy of motor performance. Sensitivity analysis and theestimated maximum machine efficiency obtained from simulationare presented.

Index Terms— Coilguns, dc motors, electromagnetic launching,energy conversion, homopolar motors, railguns.

I. INTRODUCTION

HOMOPOLAR motors and generators have been inves-tigated since their invention in the early 1800s. Much

of the work throughout the past few decades has been onlarge-scale motors for pulse power applications, the generationof high currents, energy storage mechanisms, and high-powerpropulsion drives [1]–[14]. Many successful designs haveutilized single- and multiturn coils to produce the magneticfield necessary for motor excitation, with more complicatedsuperconducting coil designs being the focus of [12]–[14].Homopolar machines may be able to reach new performancelevels as large-scale devices and as compact dc electricmotors or generators with modern high-strength permanentmagnets. The efficiency and optimum design parameters ofhomopolar machines have not been fully investigated in

Manuscript received October 12, 2014; revised December 30, 2014;accepted February 17, 2015. Date of publication March 13, 2015; date ofcurrent version May 6, 2015.

T. G. Engel is with the Department of Electrical and Computer Engi-neering, University of Missouri, Columbia, MO 65211 USA (e-mail:engelt@missouri.edu).

E. A. Kontras was with the Department of Electrical and ComputerEngineering, University of Missouri, Columbia, MO 65211 USA. He is nowwith Honeywell Federal Manufacturing and Technologies, Kansas, MO 64131USA (e-mail: ekontras@kcp.com).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPS.2015.2405531

the transient start-up phase of operation. A literature searchfound only two investigations where efficiency was reported,that of a simulated generator [10] and of an experimentalgenerator [11]. No references were found that investigatedthe effects of various parameters on homopolar efficiency.The goal of this paper is to develop a computer model ofthe homopolar motor with which to analyze the performanceand efficiency of a wide range of homopolar machines so thatefficient, low-cost, and compact homopolar machines can beeasily realized.

In general, the system resistance of a homopolar machineis quite low, which makes the homopolar motor operate atrelatively high current and low voltage [15], in contrast to morecommon rotating electrical machines. To verify the accuracyof the computer model developed in this paper, a bench-scale homopolar motor was constructed. Fig. 1(a) shows aphotograph of the homopolar motor constructed in this paperillustrating its component parts.

II. FUNDAMENTAL EQUATIONS

Fig. 1(b) shows a schematic of the homopolar motorconstructed in this paper illustrating its operation. The under-lying mechanism that generates a rotational torque in thehomopolar motor is the well-known Lorentz force. The PSpicemodel of the homopolar motor incorporates this and otherfundamental equations. The high currents in homopolar motorsgenerate Joule heating, which is of particular importance inthis paper due to its effect on system efficiency. An increas-ing conductor temperature causes a proportional resistanceincrease as a function of α, the temperature coefficient ofresistivity. The resistance of a conductor as a function oftemperature is given by

R = R0[α�T + 1]= R0 + R(T ) (1)

where R0 is the disk’s room-temperature resistance. Theresistive voltage drop can be written as

Vr = IR0 + IR(T ). (2)

Current flows through the conducting disk of Fig. 1 produc-ing a resistive voltage drop. The disk’s resistive voltage dropis found by integrating over the disk material as

Vd =∫ r1

r0

ρ I

2πrhdr

= ρ I

2π tln

(r1

r0

)(3)

0093-3813 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

1382 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 43, NO. 5, MAY 2015

Fig. 1. (a) Photograph and (b) schematic of the experimental homopolarmotor used to verify PSpice model output. Components of the homopolarmotor are labeled.

where ρ is the conductor resistivity, I is the current, h is thedisk thickness [the dimension into the drawing of Fig. 1(b)],r1 is the outer disk radius, and r0 is the inner disk radius [16].The inner and outer disk radii are determined by the regionwhere current is actually flowing in the disk.

The Lorentz force produces torque on the disk causing itto rotate. The Lorentz force expression used in this paper iswritten as [16]

F = IB� (4)

where I is the current, B is the magnetic field strength, and� is the length of conductor in the magnetic field. Equation (4)assumes that I and B are uniform over the force producingregion. The work done in moving the charge through themagnetic field region is given by the product of the force anddistance as

W = qvB� (5)

Fig. 2. Schematic showing PSpice model for calculating thermal resistance.

where q is the charge and v is the moving charge velocity.The ratio of work done and the amount of charge is anelectromotive force (emf). This emf is better known as theback voltage and is given by

Vb = vB�. (6)

The back voltage has a large impact on motor performance andis responsible for limiting its maximum angular velocity andhas a direct correlation with motor efficiency. The polarity ofthe back voltage opposes the source voltage driving the motor,and eventually a rotational velocity is reached such that theback voltage is equal to the source voltage and the currentapproaches zero [17].

The self-inductance of the circuit also creates an opposingvoltage when the current is changing with respect to time [16].The voltage is found from the basic equation for inductance as

VL = Lsd I

dt(7)

where Ls is the self-inductance of the homopolar motor.Kirchoff’s voltage law is then applied by summing the

opposing voltages and source voltage as

Vs = Vr + Vc + Vd + VL + Vb (8)

where Vs is the source voltage powering the motor and Vc isthe contact voltage drop.

III. COMPUTER MODEL

The fundamental equations (1)–(8) were used to create ananalog model of the homopolar motor in PSpice. The thermalresistance is found by computing the change in temperature fora given current input using the specific heat and mass of theconducting material. Then, using the temperature differencefrom the initial value and the temperature coefficient ofresistance, a new value for circuit resistance is found at eachtime step. The PSpice analog model to calculate the thermalresistance is shown in Fig. 2.

The individual opposing voltages in the homopolar motorcircuit were computed using (8) with the PSpice schematics

ENGEL AND KONTRAS: MODELING AND ANALYSIS OF HOMOPOLAR MOTORS AND GENERATORS 1383

Fig. 3. Schematic showing PSpice model to calculate opposing circuitvoltages.

Fig. 4. Schematic showing PSpice model to calculate opposing mechanicalforces.

shown in Fig. 3. The computation for each opposing voltageshown in Fig. 3 follows directly from (8) except for the contactresistance term. Because every brush material has differentelectrical resistance characteristics, an approximation of theincrease in electrical resistance with rotational velocity mustbe found for a given brush type. For the brush materials used intesting, a linear approximation was used, but any relationshipbetween motor speed and brush resistance could be used. Theslope and intercept of the linear approximation, representedby the variables m_contact and b_contact, respectively, arethen used to compute the contact resistance as a function ofrotational velocity. Then, multiplying by current, the opposingvoltage due to contact resistance is found.

The mechanical forces opposing motor rotation are alsoimportant to accurately model motor performance. Theseforces are modeled PSpice, as shown in Fig. 4. The force fromrotational friction is assumed constant, while the eddy-currentforce is a function of velocity. Both forces are experimentallymeasured. The opposing mechanical forces are computed andsubtracted from the Lorentz force driving the motor rotation.The Lorentz force is calculated from the current, the magneticfield strength, and the rotor disk radii, as shown in Fig. 5.The inner radius radius_0 is defined as the distance from theaxis of rotation to the inside edge of the magnetic field, and

Fig. 5. Schematic showing PSpice model to calculate the Lorentz force.

Fig. 6. Schematic showing PSpice model to calculate all the rotor motionvariables.

Fig. 7. Schematic showing PSpice model of the homopolar motor circuit.

the outer radius radius_1 is the distance to the outside edgeof the magnetic field.

The forces are added to produce an overall torque about theaxis of rotation. Dividing the total torque by the moment ofinertia yields the angular acceleration of the rotor. Integrationof the angular acceleration gives angular velocity and positionused elsewhere in the PSpice model. For convenience, therevolutions per minute (rpm) is also computed in the PSpicemodel. Fig. 6 shows the PSpice model used to calculate allthe rotor motion variables.

The circuits shown in Figs. 2–6 are connected together andused with the homopolar motor circuit to yield the completehomopolar motor model. The central homopolar motor circuitis shown in Fig. 7. The dc resistance and self-inductanceare represented in the main circuit by analog devices fromthe PSpice library. The complete homopolar motor modelcan be run for any combination of user-defined inputparameters.

The overall system efficiency of the homopolar motor iscomputed by dividing the kinetic energy of motor rotation bythe total electrical energy used in the motor. The electricalenergy is calculated by integrating electrical power, as shownin Fig. 8. Fig. 8 also shows the efficiency calculation in thePSpice model.

IV. HOMOPOLAR MOTOR CONSTRUCTION

An experimental homopolar motor was built to verify theaccuracy of the PSpice simulation. The homopolar motor

1384 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 43, NO. 5, MAY 2015

Fig. 8. Schematic showing PSpice model to calculate the electrical energyused by the homopolar motor and the overall system efficiency.

Fig. 9. Magnetic flux density between magnets in a C-shaped flux concen-trator as a function of radial position.

is shown in Fig. 1 and consists of an aluminum frame,aluminum axle with two 6-cm-diameter conducting disks, fluxconcentrators, and adjustable brushes. Only the conductingdisk drives the motor, and the rear disk measures the angularvelocity via an infrared diode sensor.

The static magnetic field for the motor is supplied byNdFeB (N52) magnets supplied by K&J Magnetics, Inc. [18].Control of the magnetic flux was done with magnetic fluxconcentrators constructed of low-carbon steel. Core lossesare not present in the homopolar motor since the magneticfield is static. Various flux concentrator designs utilizing two0.4-cm-thick and 1.3-cm-diameter magnets were modeledusing Finite Element Method Magnetics (FEMM) 4.2 [19].The simple C-shaped design was the simplest to fabricate andallowed good flux control [20]. The simple FEMM 4.2 modelwas experimentally verified using a Gauss/Tesla meter andwas found to be accurate to within 15%. The magnetic fieldslightly varies across the diameter of the magnets, as shownin Fig. 9, where the range from 0 to 13 mm represents thelength between the leading and trailing edges of the magnets.The effectiveness of the concentrator can be observed in Fig. 9,as the magnetic flux rapidly decreases to near zero outside ofthe space between the magnets.

Axle friction was reduced to the smallest possible valueusing single-point supports located at each end of the axle.The axle friction was found experimentally, and can be furtherreduced with the use of lubricants. The brushes are themain source of friction for homopolar motors, and thereis tradeoff between good electrical contact and low frictionforces [1], [21]. Regardless of brush pressure, increasing themotor velocity increases the system resistance, as there is

Fig. 10. Experimentally measured system resistance as a function of angularvelocity for various brush materials.

a separation between the brush and the conducting disk onthe order of angstroms [18]. This contact resistance is oftenreferred to as the velocity skin effect in the contact and isthought to be the primary phenomenon responsible for thevoltage drop across the contacts [22].

Different brush materials had different effects on contactresistance. Curves for system resistance as a function ofvelocity using six different brush materials were experimen-tally measured, and are shown in Fig. 10. Although exoticmetal fiber brushes have been produced that have idealcharacteristics for electric motors, tin-coated braided copperbrushes were used for the homopolar device in this paper forconvenience, cost, and ease of fabrication.

A composite disk made of circuit board with a copper layerof 0.01-mm thickness was used to pass current through themagnetic flux. A minimum thickness is ideal to reduce eddycurrents. Analytically determining or characterizing the eddycurrents in this application is beyond the scope of this paper,but the opposing force due to eddy currents was experimentallyfound as a function of velocity. The nearly linear equation foreddy-current force in newtons as a function of angular velocityin radians/second is given by

Feddy = 4 × 10−3 ω1.1912motor . (9)

The constant in (9) will increase if the conducting disk isthicker than that used in this paper.

The simplest experimental measurement was rotationalvelocity. The PSpice simulation predicted the velocity to bea maximum of 2077 r/min, using the model parameters listedin Table I. Velocity was not experimentally measured as afunction of time, and therefore, multiple maximum velocitymeasurements were taken and averaged. The average peakvelocity measured was 1860 r/min, resulting in a 10.44% errorfrom simulation. Conducting disk diameter was then decreasedfrom 10 to 8 cm, and the results were compared once more.The simulation predicted a peak velocity of 2422 r/min, andthe average maximum velocity measured was 2310 r/min,resulting in a 4.62% error.

ENGEL AND KONTRAS: MODELING AND ANALYSIS OF HOMOPOLAR MOTORS AND GENERATORS 1385

TABLE I

MODEL PARAMETERS FOR HOMOPOLAR MOTOR

Fig. 11. Effect of magnetic field strength on efficiency as predicted by thePSpice model.

V. SENSITIVITY ANALYSIS

With the accuracy of the simulation now verified withinreasonable limits, a sensitivity analysis was performed. Thesensitivity analysis is used to optimize a homopolar motorfor efficiency based on the trends observed in the simu-lation. The results given in the following are taken fromthe simulation with the homopolar motor parameters listedin Table I.

A. DC Circuit Resistance

Static resistance is critical since its value determines theLorentz driving force and Joule heating effects. Increasing thestatic resistance causes a decrease in efficiency to near-zerolevels. Decreasing the static resistance causes the motor toreach its peak efficiency. This limit occurs when the staticresistance equals the effective load resistance. With a largerstatic resistance, more energy is lost due to Joule heating. Thethermal resistance voltage drop will subsequently increase andaccount for a larger portion of the opposing voltages. It is easyto observe that a low static resistance is preferable.

B. Magnetic Field Strength

Maintaining a high flux density will increase motor perfor-mance. The efficiency over the range from 0.1 to 10 T wasanalyzed using PSpice, and the results are shown in Fig. 11where the efficiency is 1/1000 on the y-axis. There is a steadyincrease in efficiency with increasing magnetic field strength,

Fig. 12. Effect of rotor mass on efficiency as predicted by the PSpice model.

Fig. 13. Effect of conductor length on efficiency as predicted by the PSpicemodel.

which asymptotically approaches a maximum efficiencyof 0.5 at approximately 5 T. The curve nears maximumefficiency well out of the range of permanent-magnetcapabilities.

C. Mass

Peak operating efficiency slightly decreases as rotor massincreases. Mass increase also changes the shape of theefficiency curve. A small mass rotor accelerates quickly, asdoes the back voltage, and the motor reaches peak efficiencyand steady-state operation quickly. A large mass rotor cannotdevelop its back voltage as quick as a small mass rotor, whichbroadens the efficiency curve. The curve also has a lowerrate of rise and lower peak efficiency. In general, a low massrotor results in high peak efficiency for a short period of time,whereas large mass rotor results in lower peak efficiency overa longer period of time. The effect of changing rotor mass onefficiency is shown in Fig. 12, where mass is in kilograms,and efficiency is scaled 1/1000.

If high velocity is the only concern, such as in low torqueor pulse applications, a homopolar machine with low rotormass is ideal. If a large range of efficient operation isrequired, such as those in a vehicle propulsion application,a homopolar machine with a larger rotor mass is a betterchoice. The mass of the rotor is also related to the energystorage capacity of the device. Just as the energy stored

1386 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 43, NO. 5, MAY 2015

in a capacitor is proportional to the square of the voltage,for a homopolar motor, the voltage is proportional to the motorspeed and the energy stored is proportional to the square ofthe motor speed. In this sense, the homopolar motor can bethought of as a mechanical capacitor. Therefore, the mass ofthe homopolar motor is an important consideration to optimizefor different applications, as it affects not necessarily the peakefficiency, but the performance characteristics of the motor.

D. Geometry

Increasing the difference between the inner and outer diskradii lengthens the current path through the magnetic field.As this difference becomes large, the motor efficiency isgreatly improved and approaches a maximum of 0.5. Fig. 13shows the results of increasing the outer radius (m) andeffectively increasing the length of the conductor passingthrough the magnetic field. No matter the configuration, themore the length of conductor passing through the magneticfield, the more efficient and higher the performance of thehomopolar device. Unlike the broad change associated withincreasing the magnetic field, increasing the outer radius doeslittle to the overall shape of the efficiency curve. For the mostpart, changing the outer radius only increases the maximumefficiency.

VI. CONCLUSION

The transient homopolar motor simulation showedconsistent convergence to a maximum efficiency of 0.5.This is consistent with the behavior demonstrated for linearrailguns [23]. Regardless of the design changes, the homopolarmotor analyzed here is capable of operating at no more than50% efficiency, consistent with the efficiency values reportedin [10] and [11], which ranged from 0.26 to 0.48. However,design changes can drastically affect the performancecharacteristics of the homopolar motor. The simplest wayof improving efficiency is by increasing the length of theconductor passing through the magnetic field. Althoughefficiency can likewise be increased solely by increasing themagnetic field strength, this is a much more formidable taskfor permanent-magnet-based homopolar machines at valuesabove 1 T. The biggest detriments to high efficiency for thehomopolar motor are resistive heating and contact resistance,which can be minimized with proper machine design andbrush selection.

The simulation results obtained in this paper demonstratethe homopolar machine sensitivity to certain parameters andallow the user to change design variables to tailor machineperformance to a specific application. Overall and from apractical standpoint, there is still much to be learned abouthomopolar machines. Now that the maximum efficiency ofthis device has been found through simulation and verifiedexperimentally to be 0.5, the equations that characterize theefficiency of the motor are sought and will be the subject ofthe future investigations.

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