Lyall Cooper Dr. Dann ASR – D Block 10 May 2012 The Mighty Railgun (All Bark no Bite) I. Abstract The idea of this project is to build a railgun capable of firing a small metal projectile at substantial velocity. The original plan was to build iterative prototypes capable of firing small projectiles, and using them to figure out the ideal design for the final version, but the smaller versions were not very successful due to their relative lack of power, so it was difficult to use them to learn what to do. The final, largest scale railgun is powered by a bank of capacitors with an equivalent capacitance of 24.6mF charged to around 350V. This produces 3013.5J of energy discharged over approximately 58.75 µs (it varies for each firing), drawing a peak of about 1425A when firing a projectile (it once again varies) and about 1600A when not firing a projectile (short circuiting the rails). The railgun is capable of consistently firing a small piece of metal (usually aluminum or copper); however the projectile does not usually travel very far, although this is hard to measure due to the nature of the gun and the speed at which the firing takes place. II. Introduction Electricity is often seemingly mysterious, but we have come to accept and understand how through the interaction of electric and magnetic fields we can create a simple motor, as we did in the first semester. A railgun is just a linear electric motor, at very high speeds. What makes it different, however, is that it uses neither magnets nor coils of wire, and relies entirely on the induced magnetic field in the rails due to the extremely large current to produce a Lorentz Force to propel the projectile (which will be discussed in greater depth in the theory section). [2] The idea for a railgun first came from trying to figure out a way to fire a small object into space, but after some estimations and early tests, that seemed like it would be impossible given current resources (not to mention safety issues), but is still theoretically feasible, and in many ways more economical than a traditional space launch. The large currents and large forces involved create some interesting problems that need to be overcome, and leaves a lot of room to determine the best materials and design. This means that when building the various scales and iterations of the railguns, there are many things that need to be taken into account and many choices that need to be explored and made. The railgun combines precise craftsmanship with a complex physics background to make a very challenging yet interesting (if not at least dangerous) project. Firstly, the basic design of the railgun has to be sturdy enough to withstand the large forces involved, and designed in such a way that it can be safely operated. The materials have to be chosen

Figure 1: A Diagram of a simple railgun [3] to perform well in a number of categories, including conductivity, heat resistance, and overall strength and reusability. Everything must be constructed precisely enough that the projectile will always make good contact with the rails, but without hindering its movement. Capacitors have to be chosen and wired in such a way that will maximize the electrical potential energy transferred into kinetic energy while still maintaining a degree of safety and practicality. Other things, like an injection system may be required to stop the projectile from welding itself to the rails if the power is connected and the projectile is stationary between the rails. [3] This project is also very relevant to the real world, despite the fact that the final product may not be. The history of the railgun actually goes back to the mid to late 1800s, but the first proof of concept railgun wasn’t built until 1944, which could fire a 10g projectile to around 1 km/s. [4] The railgun has been continuously developed in many different places since then into what it is today, for both scientific and military use. The US Navy is in the progress of developing and testing railguns that may eventually replace traditional missiles found on naval vessels for a fraction of the cost, as the projectile is just

Figure 2: An image from the Navy's railgun test [2] a hunk of metal, as opposed to explosives and a guidance computer, among other expensive sensors and materials. [5] It is not only a weapon of the future, but also could be used for scientific and industrial testing at significantly less cost than something like a rocket sled once heat dissipation and other issues are worked out.

III. CAD Drawings (Measurements in millimeters)

Figure 4: A perspective view of the railgun

Figure 3: Front view of the railgun with measurements Figure 5: A side view of the railgun with measurements (1000 mm, 300 mm)

IV. Theory Force: As mentioned previously, a railgun works by inducing massive magnetic fields in the rails, then the current through the projectile causes a Lorentz Force to act on the projectile, propelling it down the track in the direction of the current in the right rail. The basic principle is that an electrically charged particle moving through a magnetic field experiences a force perpendicular to the direction of travel. It is this simple concept that allows a railgun to function, but the math behind it is far from simple. The magnitude of a Lorentz Force is given by the following equation: → → →

Where q is the charge of a particle, v is the velocity, and B is the magnetic field. Note how the vector cross product operator shows that the force acts perpendicularly to the direction of travel of the particle. [6] This equation can be rewritten to give the force on a wire in a magnetic field, given known current and dimensions:

Where I is the current and d is the length of the wire, or projectile in our case. The magnetic field can then be determined using the Biot–Savart Law [7], which states that the magnetic field induced by current through a wire is the following:

Where is the permeability constant, I is the current, and s is the distance from the wire. From here, we can determine the magnetic field between two wires (the two rails) using the following equation:

( )

Where d is the distance between the two wires. Then to average the magnetic field, we have to integrate (r is the radius of the wire):

∫ ∫ ( )

→ ∫ ∫ → →

→

So finally we plug that into our force equation to get the force on the projectile:

To find the total work done on the projectile, we then have to integrate again, taking into account that the current decreases exponentially as the capacitors discharge:

∫ ∫ ( ) ( )

Where R is the resistance of the entire circuit, and C is the total capacitance of the capacitors used. Note that these equations will not necessarily lead to a good approximation of how fast the projectile will go due to the friction and intense heat created by the huge current. From these equations, one can gather that the wider the projectile, the greater the force on it, but there are diminishing returns because the force is proportional to the log of the width, so there is a point where the tradeoff between width and added mass becomes a factor. Also the force is proportional to the square of the current, so doing small things to decrease the resistance of the entire system or increase the voltage of the capacitors can actually have a large effect on the theoretical power of the railgun. Heat: We know that most of the resistance of wiring and the rails dissipate electrical energy into heat. Heat is also generated by the friction of the projectile on the rails. We can make an estimate of how much heat is generated by using the following equations:

( )

∫ ( ) ∫ ( )

( ) → Since power dissipated is proportional to the resistance, it is very important to try and minimize the resistance in order to minimize the amount of energy lost to heat instead of to exerting a force on the projectile. Now if we assume we are using capacitors charged to 300 Volts with an equivalent capacitance of .01 Farads, we can make some estimates [8]:

We now know that the amount heat generated must be less than this amount, because at least some of the energy (ideally most of it) is turned into kinetic energy as opposed to heat energy. The problem is that the heat is not dissipated across all of the material as it is created so quickly, heat is mainly created at the points of contact. The ideal material then is a good conductor and has a high melting point. I chose copper for my rails because as you can see from the chart below, it has a good combination of low resistivity and high melting point.

Some possible materials [8] [9]: Material Resistivity (µΩm) Melting point (°C) Copper 1.72 1084.62 Brass (avg.) 3.9 ~920 Gold 1.62 1064.18 Steel (avg.) 17 1434 Stainless Steel 72 1434

Capacitors: The capacitors are one of the most important parts of the build, as they provide the power, and consequently the force. Simply put, a capacitor works by storing charge on parallel (or semi-parallel) plates separated by an insulator known as the dielectric. The dielectric could be air, but often things like mineral oil are used in order to increase the amount of charge that can be stored before charge jumps from one plate to the other (i.e. the capacitance). For my purposes, I need to find the right balance between voltage and capacitance, but increasing either will increase the total energy stored (dictated by the equation ⁄ as derived above). Ideally I could have a capacitor that could store an unlimited amount of charge and an infinite voltage, but of course that isn’t possible, not to mention extremely dangerous. Likely I won’t go past 400 Volts for my capacitors since I’m using an AC to DC voltage doubler/rectifier, but building a tripler or ever quadrupler is possible. Regardless, I’ll want to maximize the capacitance for whatever voltage I decide to use in order to maximize the energy stored, and therefore final kinetic energy of the projectile. I can also wire multiple capacitors in series and parallel in order to achieve the desired voltage and capacitance combination because and

. Voltage Doubler and Capacitor Bank:

Figure 6: Circuit diagram of voltage doubler and capacitor bank (lower rail is positive) The voltage doubler turns AC current into DC current while simultaneously increases the voltage by a factor of √ (the name is slightly misleading). The best analogy to describe how the doubler works is a ratcheting wrench that tightens when turned either way. On the positive part of the AC cycle, the current just goes directly to the capacitors to charge them, but on the negative cycle, the diodes act like a ratchet, preventing the built up charge from discharged while the extra capacitor (to 10µF in the circuit above) is used to build up charge that can then be “pushed” onto the capacitor bank during the negative cycle. So essentially the AC current is always pushing onto the bank during both cycles, and this is where the doubling effect comes from. The voltage is multiplied by √ instead of just 2 simply because the measured 120V from an AC socket is the RMS voltage, and not the actual peak amplitude, which is what’s being doubled.

Figure 7: The voltage doubler V. Results

Measurements: Equivalent Capacitance: The bank of 6 capacitors (3x5600µF + 2x3900µF) all in parallel has the following equivalent capacitance:

When charged with a voltage doubler connected to a regular ~120V AC outlet, the capacitors are charged to approximately 345-350V. We can then calculate the total stored energy:

Figure 8: The capacitor bank inside its protective acrylic box

The rails: Each rail is about 1m by 3cm by .75cm of pure copper, bolted down to a thick piece of wood to prevent them moving during firing due to the large force created by the interaction of the current through each rail and the induced magnetic field around each rail.

Measuring the Discharge: Firing a small piece of copper: Peak current: 1425A Average current: ~700A Discharge time: 58.75µs Shorting the rails: Peak current: 1600A Average current: ~800A Discharge time: 50µs

Measuring the attributes of how the capacitor bank discharges as a projectile is fired, such as the amount of current drawn or the time over which it took place, can be very difficult due to the nature of the railgun. One cannot simply measure the voltage drop across a known resistor to determine the current because of the sheer amount of current drawn, and because of the speed at which it happens (not to mention that it would not be very safe). So, in order to measure the discharge with something like this railgun, one must use something known as a current transformer, or CT. A CT is simply a coil of wire that is wrapped around the wire one wishes to measure the current in (it is not connected to the wire), and the changing current in the wire (in this case going from nothing to a lot) induces a magnetic field of changing strength around the wire. This changing magnetic field in turn induces a current in the CT with a voltage directly proportional to the current in the original wire. One can use a CT in conjunction with an oscilloscope to measure this induced voltage and thus measure the current through the wire. A CT may sound simple, but for their measurements to be accurate, they must be very accurately made, which for high end ones can be very expensive (the one used to measure the current in the railgun cost around $600), so one must also treat them carefully in order not to damage them in any way. Some things to note about the CT measurements: the CT used has a conversion factor of 0.1V/A, which means 1V output equals 10A through the wire being measured. However, due to the huge amount of current being drawn, a 100:1 Voltage divider was used to cut the measured output voltage of the CT by a factor of 100, so when seen on the oscilloscope, 1V equals 1000A of current through the wire.

Figure 8: The output from the CT (Voltage vs. time) when firing a small piece of copper. Each box is 1V tall by 25µs wide. The peak current is about 1425A.

Figure 9: The output from the CT (Voltage vs. time) when short circuiting the rails. Each box is 1V tall by 25µs wide. The peak current is about 1600A. Overall, the railgun was a success. It fires consistently, and the voltage doubler and capacitor bank all seem to work very well. The main problem with it, however, is that while it may fire, the projectiles do not go very far at all (usually a few meters at best), showing that there are some major inefficiencies. If it were perfectly efficient, it theoretically would be able to make a 10g projectile go about 25 m/s, but of course it is not, and while I was unfortunately unable to make any accurate measurements about projectile velocity, they were certainly lower than that, and one can safely assume that the efficiency of the gun is in the ones if not tenths of a percent. It is clear from observing a firing that a lot of energy is lost to things other than moving the projectile forwards. For example, when triggered, there is a very bright flash and very loud bang (both vary with projectile), and if one looks at where the projectile was on the rails before it was fired, there are clearly burn/melt marks from either the projectile or the rails themselves melting upon firing. All of these factors are clear example of inefficiency and lost energy that are clearly visible, and doubtless there are many more that are not as easy to see (friction, for example).

More Images:

Figure 10: The rails Figure 11: The capacitor bank and safe discharge circuit

Figure 9: A capacitor VI. The (Theoretical) Next Step With the final version complete, there are still many things that could be improved upon, or different ways of doing things that could be tried out. In my experience build this and the other versions, more power always seemed to unsurprisingly make the gun better at firing the projectile, so in the next version, I would either use more capacitors, or perhaps charge them to a higher voltage. As I’ve mentioned in previous papers, some sort of injection system (spring or CO2 powered) would undoubtedly help the firing and stop the projectile from melting as much and welding itself in place. I think I would also try to make my rails closer together, and have a larger contact area so I could place the projectile in between the rails instead of on them. This would not only help decrease resistance (minimizing melting) it should ensure that all of the force on the projectile is directly perpendicular to the rails, preventing it from prematurely losing contact from the rails and therefore not using the full amount of charge stored up in the capacitors. Finally, the projectile itself could be improved. I know with the really high powered rail guns that universities and militaries develop, plasma is used to make a connection with the rails to minimize friction and to help keep the projectile intact. While I may not be able to do something like that, I could probably at least design a projectile that would maintain contact with the rails and conduct well while still being light enough to fire well.

VII. Acknowledgements I would like to thank Dr. Dann for not only helping me a ton along the way and being lenient with deadlines, but for letting me do this crazy project in the first place. I’d also like to thank our classes three TAs, Ari Holtzman, Daniel Pugliese and Henry Bard for helping me throughout this year and especially on this project. Bibliography:

[1] “Launch to Space with an Electromagnetic Railgun”, http://research.lifeboat.com/ieee.em.pdf [2] “Railgun”, http://en.wikipedia.org/wiki/Railgun [3] “Railgun 2.0”, http://www.powerlabs.org/railgun2.htm [4] “Railgun History”, http://www.angelfire.com/fl2/quantum/history.html [5] “Navy’s new railgun test”, http://www.wired.com/dangerroom/2010/12/video-navys-mach-8-railgun- obliterates-record/ [6] “Lorentz Force”, http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html [7] “Biot-Savart Law”, http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html [8] “Copper”, http://www.wolframalpha.com/input/?i=copper [9] “Resistivity”, http://mcalc.sourceforge.net/rhoinfo.html