objective is to launch a gram-sized projectile with a velocity greater ... (velocity) state variables to be inseparably related. The ... sized projectile at 10 meters per second, currents on the order of a ... delivered power on the order of 100 kilowatts.
Session 4
uns to Break Barriers Disciplines at the United St arl E. Reinhard ABSTRACT
Engineering students tend to view their chosen discipline as isolated from other engineering disciplines while many real world problems are interdisciplinary. This paper describes how a small scale railgun is being used to change this student perception in the Department of Electrical Engineering and Computer Science of the United States Military Academy. The railgun is a simple, but elegant, device ideal for a student design project. The student must combine electrical and mechanical engineering science into a single design and analysis project; in the process, the student learns that electrical and mechanical engineering are closely related and in fact inseparable. The railgun system provides several attractive features as a teaching vehicle. The essential concepts are taught in introductory physics. The student must develop a mathematical system model (three fmt order, variable coefficient differential equations) and computer simulation to predict system behavior. The wise student makes first order engineering estimates to bound component values before trying to simulate the system. Using the simulation, the student selects an appropriate system design. The circuit is easily and inexpensively built and tested. Finally, a properly designed railgun dramatically grabs the observer's attention when fired.
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Figure 1 (a) Idealized railgtm (b) Equivalent circuit model used as an example of computers solving analytically intractablereal-world problems.
The Department of Electrical Engineering and Computer Science is using a small scale railgun at several levels of the educational process. At the design level, the railgun is one of several design projects that cadets may select in
A railgun, figure 1, is a linear machine consisting of two parallel conducting rails connected by a conducting projectile (armature); the projectile is mechanically constrained to slide down the rails. When a large current flows in the railgun, the resulting magnetic field between the rails interacts with the current to create a Lorentz for'ce that pushes the projectile down the rails. The direction of force can be determined by taking the vector cross product of the current density and the magnetic flux density. Using a capacitor as an intermediate energy storage device, the system behaves electrically much as a
their capstone electrical engineering course; the project
series resistor, inductor, and initially charged capacitor
objective is to launch a gram-sized projectile with a velocity greater than 10 meters per second. In the department's electric machine and power course, several lectures are devoted to discussing the railgun as an application of electromechanics and engineering science principles; a cadet built railgun is used to demonstrate the technology. Finally in the department's introductory computer science course, a railgun computer simulation is
circuit. Mechanically, the system is a mass with an applied force. In this system, resistance and inductance, both functions of rail and armature geometry, continuously change as the projectile moves -- causing the electrical (voltage and current) and mechanical (velocity) state variables to be inseparably related. The railgun design problem provides a conceptually simple electromechanicalsystem for students to design and build
INTRODUCTION
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t k t reinforces the interdisciplinary aspects of engineering dnd builds student confidence.
Charging Switch
Firing Switch
&AILGUN THEORY
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fvst order model of the railgun can be obtained by considering the idealized system in figure 1. In this s stem, two very wide, perfectly conducting plates are s orted by a third perfectly conducting plate with dimension h being much less than dimension d [l]. The joltage source produces a surface current density in the dpper plate given by
t
I = Kxax
(1)
daX
=
qimilar surface current densities exist in the projectile and 1 wer plate. As h is much less than d, the magnetic field &en&, H, may be considered to be approximately qniform between the conductors and can be determined by the cross product of the surface current density, K, and the unit surface normal. The resulting magnetic flux qensity, B, inside the volume bounded by the conducting shrfaces can be expressed as I
1
I
I
B = p H = p-a,x-a,, d
=
(2)
'$he magnetic flux linkage, A, of this circuit can be determined by integrating the flux density over the cross sbctional area perpendicular to the three conductors
h = P-ndS =
phxI -
S
d
(3)
m i s results in the self inductance (4)
m i s is the variable inductance shown in the equivalent ckcuit model; inductance in this simple model is a qnction of geometry and the permeability of free space. Since this is a linear, conservative system, the force on armature down the length of the rails can be qetermined by taking the derivative of the magnetic cbenergy, w,, with respect to x
Figure 2 Railgun equivalent circuit Force is a function of geometry and the current a function of x [2].
-- but not
The preceding analysis provides an attractive first order model of railgun inductance because of its simplicity. A more rigorous inductance model can be obtained by treating the rails as transmission lines.
POWER CONDITIONING Selecting a suitable power conditioning system is as important as designing the launcher. To launch a gramsized projectile at 10 meters per second, currents on the order of a kiloamp are necessary. These currents can easily be achieved by storing energy over a long period and then discharging the accumulated energy in a millisecond current pulse -- resulting very briefly in a delivered power on the order of 100 kilowatts. Capacitors are the prime candidate to meet system intermediate energy storage requirements due to their high power density relative to batteries, inductors, and electric machines; high power density allows very large, short pulse-width currents to be delivered. Capacitors are also low cost, widely available, and easily modeled. Based upon this reasoning, a capacitor bank was the preferred choice as the intermediate energy storage element. SYSTEM MODELING Considering the equivalent circuit shown in figure 2, the electrical circuit model equation is
%
V,
= IR
+
dh
dt
= IR
+
dI L-d t
+
dL I- d t (6)
I
= IR
1'
+
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dI Ldt
+
I E
E
E
dL d x I-dx d t
(7)
where Vc is capacitor voltage. Note the mechanical velocity component in the final term of this electric system equation. The equation contains three state variables: capacitor voltage, current, and velocity. This requires that two additional equations be written to uniquely solve the system. Newton's second law provides the second equation
commensurate with the launch time. Students find that narrowing the design parameter space is essential as the computer simulation required for final design does not necessarily provide insight into the effects of varying design parameters. The student will Jind ample room for creative thought in considering these and other pertinent designparameters and relationships.
SYSTEM SIMULATION
Finally, the capacitor current can be described by the equation
I =
-c-dV dt
(9)
The minus sign arises because the capacitor is delivering current. The analytical solution of these differential equations is difficult to obtain due to the variable impedance terms. Therefore, numerical techniques are necessary to solve these simultaneous equations. The model described produces an internally consistent result. The model does not, however, account for some important mechanical features of the railgun that must be considered in the design process. The rail currents interact with the magnetic flux density to deform the rails outward. The outward deformation during launch changes the contact pressure between the rails and the projectile which in turn changes the fictional force acting on the projectile and the quality of the electrical contact between the rails and projectile. Including the frictional and electrical contact behavior in the system model is not necessary. However, contact pressure and frictional forces must be considered in the mechanical design.
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PARAMETER ESTIMATION The railgun design process forces students to apply basic principles from physics, mechanics, and electrical engineering to bound the types and values of system components. For example, simple expressions relating
desired muzzle velocity, barrel length, permissible average acceleration, and launch time can be developed and applied -- which with efficiency assumptions lead to energy storage and power requirements. Rail separation is bounded by the permissible dimensions of a gram-sized projectile. The capacitor bank, rails, switch, and buswork must be selected to minimize resistance; the system resistance and capacitance must provide a time constant
A straight forward method of numerically solving these equations is a fourth-order Runge-Kutta integrator. The Runge-Kutta method will produce internally consistent results with small error. The differential equations are rewritten in the form dI dt
L
V,
-
IR
dx d t
(10)
The three state variables are given initial values. The Runge-Kutta method numerically integrates the state variable on the left side of each equation using the expression on the right-hand side. The current value for each state variable is used to calculate new values for the next iteration. The iteration process continues in small time increments until predefined conditions (elapsed time and/or projectile position) are satisfied. The Runge-Kutta integration technique is easily leaned and implemented in " m o n programming languages such as Fortran, Pascal, or C. The simulation should include a check at each iteration to insure that energy is conserved. At each instant in time, the capacitor energy, magnetic field energy, kinetic energy, and the cumulative energy resistively dissipated should sum to the energy initially stored in the capacitor. If energy balance is not within acceptable error bounds, the simulation is not programmed correctly or the timestep is too large.
CONSTRUCTION DETAILS The railgun system currently being used at the Military Academy is shown in figure 3; the system pictured is the combination of two independent student projects. The system launches a gram-sized projectile to velocities I
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Figure 3 Small scale railgun system.
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reater than 10 meters per second in the space of less than cm -- faster than the human eye can detect. Actual ystem performance matches simulated performance. i he railgun has been fired several hundred times in demonstrations.
Flectrical Svstem p e backbone of the electrical system consists of three parallel, electrolytic capacitors (Phillips model 3 186GN203S250DPA3 -- each rated 20,000 microfarads, k50 VDC). Internally, each capacitor has five conducting 6bs (to allow large currents) and an internal resistance of j milliohms at 120 Hz. The .06 farad bank capacitance Iqrovides a balance between high energy storage and a discharge time similar to the launch time. An overall equivalent circuit model for the capacitor bank was d)eterminedusing a gain phase analyzer over the range of IO0 to 40,000 Hz. Internal inductance was negligible. To date, the capacitor bank has been charged as high as 150V q!nd has produced currents as high as 6000A without qbservable performance degradation. As shown in figure 3, diodes have been connected across the capacitor tprminals to prevent damaging reverse bias voltages; €ortunately, the railgun system has always displayed vnderdamped response and no reverse bias voltages. The oharging circuit is not shown in figure 3. The other key electrical element is the closing switch; the closing switches used to date were selected for rapid turn-
on time, low resistance, and high current tolerance. Initially, a mercury vapor switch was used; however, above 2000A, the switch was prone to failure. The replacement switch selected was a phase control SCR @chardson Electronics model C290C rated 5500A one cycle surge current, 300V forward bias, and lOOA per microsecond maximum current rise rate). Testing showed that both switches had minimal effect on circuit performance. A mechanical switch is in series with the SCR as a positive safety feature; this is to prevent unintentional firing by the SCRs due to spurious voltages that may arise from test equipment.
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Mechanical Svstem The railgun itself, figure 4, consists of two copper rails separated by approximately lcm. The 3mm thick, stapleshaped aluminum projectile is approximately 1cm tall, 5mm wide, and lcm long. The railgun shown, with spring loaded rails, reflects a design improvement dictated by experience with a previous design having two fixed rails. With fixed rails, each projectile must be manufactured to precise tolerances for correct pressure between the rails and projectile. Too little pressure results in poor electrical contact and no current. Excessive pressure causes a frictional force preventing the projectile's movement -- in this event, ohmic heating welds the projectile to the rails. The spring loaded rails consistently provide the correct pressure. The kiloamp currents that flowed during each launch caused significant I
1'
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traveled divided by the time calculated for the projectile to drop a known distance.
CONCLUSION The use of a small scale railgun has improved the instruction of undergraduate engineers at the United States Military Academy by presenting a simple interdisciplinary design problem to cadets. Designing this system helps cadets see and understand the importance of multiple disciplines in solving a real-world problem. The project requires the student to exercise the spectrum of his engineering skills: modeling the physical system, estimating reasonable component values, and applying numerical techniques to solve a system of differential equations. The railgun is easily and inexpensively constructed. Finally, the student must measure the performance of the system.
ACKNOWLEDGMENTS
damage to the rails and projectile; the damage in firing the projectile demonstrated the importance of rail and projectile material selection. The rail material must be very conductive, hard, and have a high melting temperature. The issue of material selection has been a valuable real-world learning point.
The railgun power supply system pictured was designed and tested by Peter Luhowy who graduated from the United States Military Academy in 1993. The railgun pictured was designed and tested by Chong Yim, USMA Class of 1995. Craig Scott Smith, USMA Class of 1995, demonstrated both velocity measurement techniques in his coilgun project. Major John E. Post provided early encouragement and editorial assistance in preparing this paper.
SYSTEM PERFORMANCE MEASUREMENTS
REFERENCES
The principal electrical measurement tool was a digital oscilloscope to record the changing capacitor voltage. The oscilloscope was set to pretrigger as the voltage dropped slightly below the charged value. The capacitor voltage traces have compared favorably (but not exactly) to the voltage traces predicted by simulation. The current trace can be determined from the product of capacitance and the slope of the voltage trace.
[l] Herbert H. Woodson and James R. Melcher, Electromechanical Dvnamics Part I, (Malabar, Florida: Robert Krieger Publishing 1990), 322-323. Idealized railgun system based upon problem 6.12.
Two approaches have been taken in measuring projectile velocity. First, a VCR camera recorded projectile movement with a linear scale and stopwatch within the field of view; velocity being determined by displacement as a function of time from frame to frame. Unfortunately, the frame rate of a standard VCR camera is too slow to measure a projectile moving at ten plus meters per second. As a fallback, projectile velocity was determined ballistically. The time required for the projectile to drop a known distance is constant. Neglecting resistive air effects, the average velocity is the horizontal distance
Karl E. Reinhard is currently a Major on active duty in the U.S. Army and is serving as an Assistant Professor in the Department of Electrical Engineering and Computer Science. He received the BS degree in 1982 from the U.S. Military Academy and was commissioned in the Ordnance Corps. He earned the MS degree from the University of Texas-Austin in 1992 and joined the Department of Electrical Engineering and Computer Science. He is responsible for the department's electric power courses. He is a licensed professional engineer.
Figure 4
Student designed railgun
[2] Karl E. Reinhard, "A Methodology for Selecting an Electromagnetic Gun System," ( M S Thesis, University of Texas at Austin, 1992), 17-21.
***** Karl E. Reinhard
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