IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 43, NO.5, MAY 2015 1381 Modeling and Analysis of Homopolar Motors and Generators Thomas G. Engel, Senior Member, IEEE, and Evan A. Kontras, Graduate Student Member, IEEE Abstract— The dc homopolar motor converts electrical energy into mechanical rotational energy using the Lorentz force. The same machine can be operated in reverse to convert mechanical energy into dc electrical energy. To better understand homopolar motors and their suitability for use in various applications, a computer model was created using PSpice. Forces opposing the motor rotation include back voltage, eddy currents, moment of inertia, and sliding contact friction and are analyzed in detail. Forces and torques are discussed and calculated analytically. The capabilities of the homopolar machine operating as both a motor and a generator are considered. Using current research from the University of Missouri on helical guns and railguns, which utilize similar electromagnetic forces for linear acceleration, the maximum efficiency of the homopolar motor during transient start-up phase is examined. The measured homopolar motor efficiency in this paper asymptotically approaches 50% and is determined by several variables. Experimental data are collected and used to compare the simulation results and verify the accuracy of motor performance. Sensitivity analysis and the estimated maximum machine efficiency obtained from simulation are presented. the transient start-up phase of operation. A literature search found only two investigations where efficiency was reported, that of a simulated generator [10] and of an experimental generator [11]. No references were found that investigated the effects of various parameters on homopolar efficiency. The goal of this paper is to develop a computer model of the homopolar motor with which to analyze the performance and efficiency of a wide range of homopolar machines so that efficient, low-cost, and compact homopolar machines can be easily realized. In general, the system resistance of a homopolar machine is quite low, which makes the homopolar motor operate at relatively high current and low voltage [15], in contrast to more common rotating electrical machines. To verify the accuracy of the computer model developed in this paper, a benchscale homopolar motor was constructed. Fig. 1(a) shows a photograph of the homopolar motor constructed in this paper illustrating its component parts. Index Terms— Coilguns, dc motors, electromagnetic launching, energy conversion, homopolar motors, railguns. II. F UNDAMENTAL E QUATIONS I. I NTRODUCTION H OMOPOLAR motors and generators have been investigated since their invention in the early 1800s. Much of the work throughout the past few decades has been on large-scale motors for pulse power applications, the generation of high currents, energy storage mechanisms, and high-power propulsion drives [1]–[14]. Many successful designs have utilized single- and multiturn coils to produce the magnetic field necessary for motor excitation, with more complicated superconducting coil designs being the focus of [12]–[14]. Homopolar machines may be able to reach new performance levels as large-scale devices and as compact dc electric motors or generators with modern high-strength permanent magnets. The efficiency and optimum design parameters of homopolar 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 of current version May 6, 2015. T. G. Engel is with the Department of Electrical and Computer Engineering, University of Missouri, Columbia, MO 65211 USA (e-mail:
[email protected]). E. A. Kontras was with the Department of Electrical and Computer Engineering, University of Missouri, Columbia, MO 65211 USA. He is now with Honeywell Federal Manufacturing and Technologies, Kansas, MO 64131 USA (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPS.2015.2405531 Fig. 1(b) shows a schematic of the homopolar motor constructed in this paper illustrating its operation. The underlying mechanism that generates a rotational torque in the homopolar motor is the well-known Lorentz force. The PSpice model of the homopolar motor incorporates this and other fundamental equations. The high currents in homopolar motors generate Joule heating, which is of particular importance in this paper due to its effect on system efficiency. An increasing conductor temperature causes a proportional resistance increase as a function of α, the temperature coefficient of resistivity. The resistance of a conductor as a function of temperature is given by R = R0 [αT + 1] = R0 + R(T ) (1) where R0 is the disk’s room-temperature resistance. The resistive voltage drop can be written as Vr = IR0 + IR(T ). (2) Current flows through the conducting disk of Fig. 1 producing a resistive voltage drop. The disk’s resistive voltage drop is found by integrating over the disk material as r1 ρI dr Vd = r0 2πr h ρI r1 = ln (3) 2πt r0 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. (6) The back voltage has a large impact on motor performance and is responsible for limiting its maximum angular velocity and has a direct correlation with motor efficiency. B is the magnetic field strength. (a) Photograph and (b) schematic of the experimental homopolar motor used to verify PSpice model output.1382 IEEE TRANSACTIONS ON PLASMA SCIENCE. Kirchoff’s voltage law is then applied by summing the opposing voltages and source voltage as VL = L s Fig. The ratio of work done and the amount of charge is an electromotive force (emf). 5. 43. The inner and outer disk radii are determined by the region where current is actually flowing in the disk. and eventually a rotational velocity is reached such that the back voltage is equal to the source voltage and the current approaches zero [17]. NO. The individual opposing voltages in the homopolar motor circuit were computed using (8) with the PSpice schematics . Schematic showing PSpice model for calculating thermal resistance. where q is the charge and v is the moving charge velocity. Equation (4) assumes that I and B are uniform over the force producing region. The self-inductance of the circuit also creates an opposing voltage when the current is changing with respect to time [16]. Then. where ρ is the conductor resistivity. r1 is the outer disk radius. Components of the homopolar motor are labeled. The Lorentz force expression used in this paper is written as [16] F = IB (4) where I is the current. and r0 is the inner disk radius [16]. C OMPUTER M ODEL The fundamental equations (1)–(8) were used to create an analog model of the homopolar motor in PSpice. The voltage is found from the basic equation for inductance as dI (7) dt where L s is the self-inductance of the homopolar motor. The thermal resistance is found by computing the change in temperature for a given current input using the specific heat and mass of the conducting material. using the temperature difference from the initial value and the temperature coefficient of resistance. h is the disk thickness [the dimension into the drawing of Fig. VOL. III. 1(b)]. The Lorentz force produces torque on the disk causing it to rotate. MAY 2015 Fig. 2. 2. This emf is better known as the back voltage and is given by Vb = vB. The PSpice analog model to calculate the thermal resistance is shown in Fig. a new value for circuit resistance is found at each time step. 1. The polarity of the back voltage opposes the source voltage driving the motor. I is the current. and is the length of conductor in the magnetic field. The work done in moving the charge through the magnetic field region is given by the product of the force and distance as W = qvB (5) Vs = Vr + Vc + Vd + VL + Vb (8) where Vs is the source voltage powering the motor and Vc is the contact voltage drop. 8. The Lorentz force is calculated from the current. 7. forces. For the brush materials used in testing. as shown in Fig. the outer radius radius_1 is the distance to the outside edge of the magnetic field. Fig. the magnetic field strength. Then. For convenience. as shown in Fig. 6. 6 shows the PSpice model used to calculate all the rotor motion variables. represented by the variables m_contact and b_contact. as shown in Fig. 5. 5. 8 also shows the efficiency calculation in the PSpice model. The inner radius radius_0 is defined as the distance from the axis of rotation to the inside edge of the magnetic field. Schematic showing PSpice model to calculate opposing mechanical shown in Fig. Fig. The mechanical forces opposing motor rotation are also important to accurately model motor performance. The opposing mechanical forces are computed and subtracted from the Lorentz force driving the motor rotation. Schematic showing PSpice model to calculate all the rotor motion variables.ENGEL AND KONTRAS: MODELING AND ANALYSIS OF HOMOPOLAR MOTORS AND GENERATORS Fig. Fig. The force from rotational friction is assumed constant. 4. 3 follows directly from (8) except for the contact resistance term. 7. 3. Schematic showing PSpice model to calculate opposing circuit voltages. IV. a linear approximation was used. Dividing the total torque by the moment of inertia yields the angular acceleration of the rotor. while the eddy-current force is a function of velocity. The electrical energy is calculated by integrating electrical power. The forces are added to produce an overall torque about the axis of rotation. Fig. Both forces are experimentally measured. are then used to compute the contact resistance as a function of rotational velocity. respectively. The computation for each opposing voltage shown in Fig. Fig. The circuits shown in Figs. 1383 Schematic showing PSpice model to calculate the Lorentz force. the opposing voltage due to contact resistance is found. the revolutions per minute (rpm) is also computed in the PSpice model. 3. and Schematic showing PSpice model of the homopolar motor circuit. and the rotor disk radii. Because every brush material has different electrical resistance characteristics. The homopolar motor . The slope and intercept of the linear approximation. The complete homopolar motor model can be run for any combination of user-defined input parameters. The dc resistance and self-inductance are represented in the main circuit by analog devices from the PSpice library. 2–6 are connected together and used with the homopolar motor circuit to yield the complete homopolar motor model. The central homopolar motor circuit is shown in Fig. but any relationship between motor speed and brush resistance could be used. Fig. an approximation of the increase in electrical resistance with rotational velocity must be found for a given brush type. multiplying by current. The overall system efficiency of the homopolar motor is computed by dividing the kinetic energy of motor rotation by the total electrical energy used in the motor. H OMOPOLAR M OTOR C ONSTRUCTION An experimental homopolar motor was built to verify the accuracy of the PSpice simulation. Integration of the angular acceleration gives angular velocity and position used elsewhere in the PSpice model. 4. These forces are modeled PSpice. The simple FEMM 4.44% error from simulation. Various flux concentrator designs utilizing two 0.1384 IEEE TRANSACTIONS ON PLASMA SCIENCE. and ease of fabrication.62% error. and therefore.2 model was experimentally verified using a Gauss/Tesla meter and was found to be accurate to within 15%. Fig. Fig. and the rear disk measures the angular velocity via an infrared diode sensor. MAY 2015 Fig. as there is a separation between the brush and the conducting disk on the order of angstroms [18]. multiple maximum velocity measurements were taken and averaged. Although exotic metal fiber brushes have been produced that have ideal characteristics for electric motors. and can be further reduced with the use of lubricants. increasing the motor velocity increases the system resistance. The nearly linear equation for eddy-current force in newtons as a function of angular velocity in radians/second is given by 1. The simulation predicted a peak velocity of 2422 r/min. Control of the magnetic flux was done with magnetic flux concentrators constructed of low-carbon steel.4-cm-thick and 1. 9. 1 and consists of an aluminum frame. The simplest experimental measurement was rotational velocity. [21]. 9. Different brush materials had different effects on contact resistance. Analytically determining or characterizing the eddy currents in this application is beyond the scope of this paper. The effectiveness of the concentrator can be observed in Fig. as the magnetic flux rapidly decreases to near zero outside of the space between the magnets.3-cm-diameter magnets were modeled using Finite Element Method Magnetics (FEMM) 4. and the results were compared once more. Feddy = 4 × 10−3 ωmotor (9) The constant in (9) will increase if the conducting disk is thicker than that used in this paper.01-mm thickness was used to pass current through the magnetic flux. and are shown in Fig. Only the conducting disk drives the motor. . Axle friction was reduced to the smallest possible value using single-point supports located at each end of the axle. Experimentally measured system resistance as a function of angular velocity for various brush materials. Regardless of brush pressure. where the range from 0 to 13 mm represents the length between the leading and trailing edges of the magnets. using the model parameters listed in Table I. The axle friction was found experimentally. as shown in Fig. This contact resistance is often referred to as the velocity skin effect in the contact and is thought to be the primary phenomenon responsible for the voltage drop across the contacts [22]. The static magnetic field for the motor is supplied by NdFeB (N52) magnets supplied by K&J Magnetics. resulting in a 4. The brushes are the main source of friction for homopolar motors. [18]. 8. cost. 5. Schematic showing PSpice model to calculate the electrical energy used by the homopolar motor and the overall system efficiency. A minimum thickness is ideal to reduce eddy currents. The average peak velocity measured was 1860 r/min.1912 . and adjustable brushes. Core losses are not present in the homopolar motor since the magnetic field is static. tin-coated braided copper brushes were used for the homopolar device in this paper for convenience. and the average maximum velocity measured was 2310 r/min. Inc. The simple C-shaped design was the simplest to fabricate and allowed good flux control [20]. Magnetic flux density between magnets in a C-shaped flux concentrator as a function of radial position. Velocity was not experimentally measured as a function of time. The magnetic field slightly varies across the diameter of the magnets.2 [19]. VOL. NO. 9. is shown in Fig. A composite disk made of circuit board with a copper layer of 0. resulting in a 10. Curves for system resistance as a function of velocity using six different brush materials were experimentally measured. aluminum axle with two 6-cm-diameter conducting disks. 10. flux concentrators. Conducting disk diameter was then decreased from 10 to 8 cm. but the opposing force due to eddy currents was experimentally found as a function of velocity. 10. and there is tradeoff between good electrical contact and low friction forces [1]. 43. The PSpice simulation predicted the velocity to be a maximum of 2077 r/min. 11 where the efficiency is 1/1000 on the y-axis. B. The mass of the rotor is also related to the energy storage capacity of the device. 13. The efficiency over the range from 0. where mass is in kilograms. In general. A large mass rotor cannot develop its back voltage as quick as a small mass rotor. S ENSITIVITY A NALYSIS With the accuracy of the simulation now verified within reasonable limits. Mass Peak operating efficiency slightly decreases as rotor mass increases. Fig. If a large range of efficient operation is required. Just as the energy stored .5 at approximately 5 T. The curve nears maximum efficiency well out of the range of permanent-magnet capabilities. more energy is lost due to Joule heating. The effect of changing rotor mass on efficiency is shown in Fig.1 to 10 T was analyzed using PSpice. DC Circuit Resistance Static resistance is critical since its value determines the Lorentz driving force and Joule heating effects. a homopolar machine with a larger rotor mass is a better choice. C. and efficiency is scaled 1/1000. a low mass rotor results in high peak efficiency for a short period of time. With a larger static resistance.ENGEL AND KONTRAS: MODELING AND ANALYSIS OF HOMOPOLAR MOTORS AND GENERATORS 1385 TABLE I M ODEL PARAMETERS FOR H OMOPOLAR M OTOR Fig. A small mass rotor accelerates quickly. 12. Fig. Increasing the static resistance causes a decrease in efficiency to near-zero levels. Effect of rotor mass on efficiency as predicted by the PSpice model. a sensitivity analysis was performed. whereas large mass rotor results in lower peak efficiency over a longer period of time. The results given in the following are taken from the simulation with the homopolar motor parameters listed in Table I. such as those in a vehicle propulsion application. The sensitivity analysis is used to optimize a homopolar motor for efficiency based on the trends observed in the simulation. Magnetic Field Strength Maintaining a high flux density will increase motor performance. as does the back voltage. Mass increase also changes the shape of the efficiency curve. There is a steady increase in efficiency with increasing magnetic field strength. 12. If high velocity is the only concern. It is easy to observe that a low static resistance is preferable. A. The thermal resistance voltage drop will subsequently increase and account for a larger portion of the opposing voltages. The curve also has a lower rate of rise and lower peak efficiency. 11. This limit occurs when the static resistance equals the effective load resistance. and the motor reaches peak efficiency and steady-state operation quickly. Effect of conductor length on efficiency as predicted by the PSpice model. a homopolar machine with low rotor mass is ideal. Effect of magnetic field strength on efficiency as predicted by the PSpice model. Decreasing the static resistance causes the motor to reach its peak efficiency. and the results are shown in Fig. V. which broadens the efficiency curve. which asymptotically approaches a maximum efficiency of 0. such as in low torque or pulse applications. “Composite solid armature consolidation by pulse power processing: A novel homopolar generator application in EML technology. 20.” Galilean Electrodyn. Magn.” IEEE Trans. 2001. J. 1989. Engel. Mazzoni.. D. Shopen. 107–108. [13] B. D. and J. 7. changing the outer radius only increases the maximum efficiency. H.” IEEE Trans. C. Magn. Chabay and B. New York. vol. D. USA: Wiley. A. C. J. “The high voltage homopolar generator. N. Schaubel. M. 4.” IEEE Trans. Mar. no.kjmagnetics. Guala-Valverde. D. USA: Macmillan. H. IEEE Int. 13 shows the results of increasing the outer radius (m) and effectively increasing the length of the conductor passing through the magnetic field.. Therefore. F. Hamilton. Rylander. The simulation results obtained in this paper demonstrate the homopolar machine sensitivity to certain parameters and allow the user to change design variables to tailor machine performance to a specific application. Although efficiency can likewise be increased solely by increasing the magnetic field strength. Unlike the broad change associated with increasing the magnetic field. Conf. A. no. Argentina. 1997. 1989.com/ [19] D. and D. R. no.femm.” IEEE Trans. 1690–1694. Jan.. no. “Balcones homopolar generator power supply. no. the mass of the homopolar motor is an important consideration to optimize for different applications. Jun. Hayes and R. [21] J. The biggest detriments to high efficiency for the homopolar motor are resistive heating and contact resistance. 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