1 ELEC 3105 Basic EM and Power EngineeringLinear motors: Case 1 no load Case 2 with load: Power consumption Linear Motor / Generator Alright class, tomorrow is your E&M exam You may bring a formula sheet, anything on the formula sheet may be used on the test. Next day
2 ELEC 3105 Basic EM and Power EngineeringLinear motors From this To these
3 Linear motor Assume external applied B field much greater than B field generated by current in bar. Metal fixed rail Magnetic flux density into page External applied field Movable metal bar
4 Linear motor Moves bar towards the right increasing the size of the current loop. Load force restricts movement of sliding bar Metal fixed rail Current limiting resistor
5 Linear motor By Lenz’s law induced magnetic field must be such as to oppose the increasing flux. Increasing flux due to expanding loop This is accomplished by inducing a magnetic field in the opposite direction to the applied magnetic field.
6 Linear motor The induced current required to produce the opposing magnetic field has a direction opposite to the current supplied from the battery. The induced emf opposing the battery voltage is known as the “BACK EMF”
7 Linear motor Work done on positive charge in moving from bottom to top of bar. ++++ Work per unit charge ---- By definition “back emf”given by: Charge moves along with bar
8 Linear motor Equivalent circuit Bar
9 Linear motor Linear motorAlternate approach to obtaining expression for back emf. Metal fixed rail In time loop increases in area by:
10 Linear motor Linear motorAlternate approach to obtaining expression for back emf. Metal fixed rail Change in flux over time interval is:
11 Linear motor Linear motorAlternate approach to obtaining expression for back emf. Metal fixed rail From Faraday’s law: Minus sign reflects Lenz’s law, induced opposes change in flux.
12 ELEC 3105 Basic EM and Power EngineeringLinear motors: Case 1 no load
13 Linear motor: Case 1 Suppose there is no load force.Once battery is connected the bar will accelerate to the right until: Metal fixed rail Once this speed is reached I = 0 and there is no force to accelerate the bar further. Movable metal bar can slip without friction, …..
14 Linear motor Case 1 Equivalent circuit Bar Then with no load force.From general expression of emf Then with no load force.
15 ELEC 3105 Basic EM and Power EngineeringCase 2 with load
16 Linear motor: Case 2 Suppose there is a load force.Once battery is connected the bar will accelerate to the right until: Metal fixed rail Once this speed is reached the bar no longer accelerates and Movable metal bar can slip with friction, …..
17 Linear motor Case 2 Equivalent circuit From balanced forces Barlinear relation between
18 Linear motor Case 2 Load / speed characteristic plot linear relation
19 Linear motor Case 2 Load / speed characteristic plot MotorInternal combustion engine can’t operate at zero speed. Need transmission, clutch, … DC motor does not need a transmission – major saving in weight, assembly, …. Motor Maximum current is drawn at startup (may need to take action to protect motor windings from overheating) A stalled motor burns faster Maximum force (or torque) is obtained at zero speed
20 ELEC 3105 Basic EM and Power EngineeringSTART Power consumption
21 Linear motor Power Mechanical power deliveredElectrical power consumed Metal fixed rail For conservation of energy
22 Linear motor Power Mechanical power deliveredElectrical power consumed For conservation of energy Recall Then
23 ELEC 3105 Basic EM and Power EngineeringSTART Linear Motor / Generator
24 Linear motor / GeneratorSuppose direction of load force is as shown. We pull on the bar. Now Since magnetic and mechanical forces act in same direction.
25 Linear motor / GeneratorLoad / speed characteristic plot Current flows in a direction to charge the battery. Generator Motor
26 Linear motor / GeneratorSuppose we reverse the direction of load force. We pull on the bar. Decreasing flux results in reverse polarity of back emf. Now back emf and battery add giving larger current. System acts as a generator Application in electric cars: regenerative braking: motor can be used to recharge battery during braking.
27 Linear motor / GeneratorLoad / speed characteristic plot Back emf and battery polarities combine producing larger current. Current flows in a direction to charge the battery. Generator Generator Motor
28 Linear motor / GeneratorAny mechanical force on the bar will induce the bar to move in that direction. Suppose now we remove the battery altogether The load force pulling on the bar will generate a current in the loop and as such the system acts as a generator for all load forces applied. The direction of the current is determined by the direction of the load force.
29 Linear Generator Load / speed characteristic plot No battery in systemCurrent flows in CW through resistor. Current flows in CCW through resistor. Generator Generator
30 ELEC 3105 Basic EM and Power EngineeringThe Big Bad Rail Gun
31 The Big Bad Rail Gun Welcome to the newly redesigned haven for the rail gun enthusiast. This page covers some of the latest techniques in electromagnetic propulsion, but the construction of a rail gun is a perilous undertaking so use the information contained herein at your own risk. NOTE: In the interest of simplicity vector directions are ignored, it is assumed that the magnitude is as calculated and in the desired direction. What is a rail gun? A rail gun in it's simplest form is a pair of conducting rails separated by a distance L and with one rail connected to the positive and one the negative side of a power source supplying voltage V and current I. A conducting projectile bridges the gap L between the rails, completing the electrical circuit. As current I flows through the rails, a magnetic field B is generated with an orientation dictated by the right hand rule and with a magnitude governed by equation 1.
32 The Big Bad Rail Gun (1) B=NuI B = Magnetic field strength (Teslas)N = Number of turns in solenoid (1 in our case) u = 1.26x10^-6 (The magnetic permeability of free space, Henries/Meter) I = Current through rails and projectile (Amperes) Figure 1: Simple Rail Gun
33 The Big Bad Rail Gun When a current I moves through a conductor of length L in the presence of a magnetic field B, the conductor experiences a force F according to equation 2. (2) F = ILB F = Force on conductor (projectile, in Newtons) I = Current through rails and projectile (Amperes) L = Length of rail separation (Meters) B = Magnetic field strength (Teslas) The direction of the force depends on the direction of the current through the projectile and the magnetic field since the force is truly a vector with direction dictated by the cross product of the vector quantities I and B. In Figure 1, the force is oriented down the rails, away from the power source. See the section on the Right Hand Rule below for a detailed description of this.
34 The Big Bad Rail Gun How fast does a rail gun projectile go?Short answer: currently about 4 km/s The speed of a rail gun slug is determined by several factors; the applied force, the amount of time that force is applied, and friction. Friction will be ignored in this discussion, as it's effects can only be determined through testing. If this concerns you, assume a friction force equal to 25% of driving force. The projectile, experiencing a net force as described in the above section, will accelerate in the direction of that force as in equation 3. (3) a = F/m a = Acceleration (Meters/second^2) F = Force on projectile (Newtons) m = Mass of projectile (Kilograms)
35 The Big Bad Rail Gun Unfortunately, as the projectile moves, the magnetic flux through the circuit is increasing and thus induces a back EMF (Electro Magnetic Field) manifested as a decrease in voltage across the rails. The theoretical terminal velocity of the projectile is thus the point where the induced EMF has the same magnitude as the power source voltage, completely canceling it out. Equation 4 shows the equation for the magnetic flux. (4) H = BA H = Magnetic Flux (Teslas x Meter^2) B = Magnetic field strength (Teslas) (Assuming uniform field) A = Area (Meter^2)
36 The Big Bad Rail Gun Equation 5 shows how the induced voltage V(i) is related to H and the velocity of the projectile. (5) V(i) = dH / dt = BdA / dt = BLdx / dt V(i) = Induced voltage dH / dt = Time rate of change in magnetic flux B = Magnetic field strength (Teslas) dA / dt = Time rate of change in area L = Width of rails (Meters) dx / dt = Time rate of change in position (velocity of projectile) Since the projectile will continue to accelerate until the induced voltage is equal to the applied, Equation 6 shows the terminal velocity v(max) of the projectile. (6) v(max) = V / (BL) v(max) = Terminal velocity of projectile (Meters/second) V = Power source voltage (Volts)
37 The Big Bad Rail Gun These calculations give an idea of the theoretical maximum velocity of a rail gun projectile, but the actual muzzle velocity is dictated by the length of the rails. The length of the rails governs how long the projectile experiences the applied force and thus how long it gets to accelerate. Assuming a constant force and thus a constant acceleration, the muzzle velocity (assuming the projectile is initially at rest) can be found using Equation 7. (7) v(muz) = (2DF / m)^.5 = (2DILB / m)^.5 = I(2DLu / m)^.5 v(muz) = Muzzle velocity (Meters/Second) D = Length of rails (Meters) F = Force applied (Newtons) m = Mass of projectile (Kilograms) I = Current through projectile (Amperes) L = Width between rails (Meters) B = Magnetic field strength (Teslas) u = 1.26x10^-6 (The magnetic permeability of free space, Henries/Meter) These calculations ignore friction and air drag, which can be formidable at the speeds and forces applied to the rail gun slug. Top rail gun designs currently can launch a 2kg projectile with a muzzle velocity of close to 4km/s on roughly 6 meter rails. To reach this kind of velocity, the power source must provide roughly 6.5 million Amps. Ouch.
38 The Big Bad Rail Gun What is the right hand rule?The right hand rule is a mnemonic for memorizing the orientation of fields, forces, or other vector quantities after the cross product of two vector quantities is taken. For example, the direction of the magnetic field around a conductor due to a current can be determined by pointing the right thumb in the direction of the current and curling the other four fingers as if grasping the conductor. The magnetic field similarly exists as a vector field circling the conductor in the direction indicated by your fingers. See Figure 2 for a visual representation. Figure 2: Right hand rule for magnetic field from current through conductor
39 The Big Bad Rail Gun In addition, the right hand rule comes into play when performing cross products of vector quantities. For example, when figuring out which way the projectile in a rail gun will go, you look to Equation 2. Equation 2 is truly a cross product, but presented as a simple multiplication for the sake of simplicity. The force exerted on the projectile is the cross product of scalar length L, vector i the path of the current in the projectile, and vector field B the magnetic field. When determining the direction of this force we can use the right hand rule. Since all the angles involved are 90 degrees, the resultant force has a magnitude resulting from the simple multiplication of the magnitude of i and B and the value of L. (|F|=L|i||B|) To determine the direction, lay your right hand along the path of the current through the projectile, with your fingers pointing in the direction the current is traveling. Next, curl your fingers in the direction of the B field. Your thumb will now be pointing in the direction of the applied force. See Figure 3 for a visual representation. Figure 3: Right hand rule for cross product
40 What should the rails be made of?The Big Bad Rail Gun What should the rails be made of? The rails can be made of any conductive material. However, the best rail material will depend on your specific design. Important characteristics of a good rail material are high conductivity, high strength, high machinability, resistance to corrosion, high melting point, availability, compatibility with slug material, and finally price. Clearly, the choice of rail materials will be a compromise. A good place to start is electrical conductivity. The amount of heat the rails will need to withstand will be in large part dependent on their resistance. Below is a list of several metals and their conductivities, melting points, and heat capacity.
41 The Big Bad Rail Gun Material Resistivity(Ohm/cm) Melting Point (C) Heat Capacity (J/g°C) CMW® D158F n/a Silver CP Grade Oxygen Free Copper 1050-O Aluminum 440-A Stainless Steel Haynes 25 Super Alloy Titanium 6-4 Annealed Tungsten
42 The Big Bad Rail Gun Clearly there is a wide range of options. Titanium will absorb 100 times as much energy from a given current when compared to CP grade (pure) silver. Tungsten melts at a temperature roughly 5 times higher than aluminum, but aluminum will take over 6 times more energy per gram to heat. Thus, all other things being equal, tungsten will melt from a lower input of energy than aluminum. This is a little misleading, since the heat capacity also is dependent on mass and tungsten is about 3-4 times as dense as aluminum. Thus the above comparison would involve a smaller sample of tungsten (3-4 times smaller by volume) than the aluminum sample. The ideal rail combines the strengths of several materials, a metal composite with each metal performing the function it excels at. For example, the contact surfaces would have high conductivity, low friction, and high melting point; the support structure would be strong, light, and conduct heat away quickly. A good rail could be made of 7075 T-6 aluminum supporting a CMW® D158F silver and graphite low friction contact surface. A pair of rails fabricated in this way could be combined in a carbon fiber barrel assay for increased stiffness and weight savings. The strength and stiffness of the rail is of utmost importance, as the forces generated on the projectile are also felt by the rails.
43 The Big Bad Rail Gun How should the rails be shaped?The primary purpose of the rails in a rail gun is to conduct electricity to the projectile, build a magnetic field, and guide the projectile out of the device. The actual shape of the rail is important in two instances. 1) The contact patch between rail and projectile. 2) The structural rigidity of the rail in the horizontal plane (in plane with but perpendicular to projectile motion). Obviously, the rail surface in contact with the projectile should have the same contour as the projectile itself. The best shape for this interface is flat. This shape is optimal because it is cheaper to make, simplifies the production of high-precision surfaces, facilitates rail compositing as mentioned above, and minimizes match-up errors between projectile and rail. Use of a grooved rail and projectile could be performed, with the benefit of increased contact area and thus a reduction in current density, but in practice, producing these parts would be too costly. Considering the lifespan of rails (around 100 shots, give or take 100), the simplicity of the rail is imperative. The shape of the rail not in contact with the projectile must be designed for the utmost in structural rigidity. Fortunately the vast bulk of force experienced by the rails is in one direction, a result of the same equation that dictates the forward motion of the projectile. Thus the rail must be designed to withstand an outward force equal to that exerted on the projectile. As you can imagine this force is huge, as it tends to send the projectile out of the gun at a few km/s.
44 The Big Bad Rail Gun So, borrowing a few lessons from structural engineering, an ideal rail support (structural backing for contact/conducting surface) would resemble a beam or truss designed for maximum resistance to bending. The cross section of this beam can take on many shapes based on the resources at hand, I or H beam, U channeled, closed rectangular, closed triangular, and so forth. If you don't already have a bunch of steel girders laying around, I'd get our the old mechanics textbook and calculate the bending moments for some different cross sections and determine which you'll use. The complexity, the strength to weight of the cross-section, availability, and ultimately cost will determine your choice of rail. How stiff do the rails need to be? The rails need to be stiff enough that at maximum force, they do not deflect enough to break the circuit of rails and projectile. The tolerances of your design, sabot geometry, and even pulse shape will determine the structural requirements. You may want to design in some interference to account for the spreading of the rails, build in some toe-in, design a suspension system to maintain solid contact, or design the sabot to ride on top of the rails rather than between them. This will allow you to account for rail movement at the expense of increased projectile mass, difficulty in maintaining optimal current density, and so forth. As you may have noticed, rail guns are a pile of compromises. It is definitely no easy task to trick the forces of nature into throwing a 2 kg piece of tungsten around at 4 km/s.
45 The Big Bad Rail Gun Working from the example given earlier (2 kg projectile, 4 km/s muzzle velocity, 6 m rail length) the forces present are as follows. Assuming constant acceleration, the time the projectile spends in the rails is: (1) T = d / va t = time spent in contact with rails, (seconds) d = distance traveled, length of rails, .006 km (kilometers) va = average velocity, (vm-vo)/2, muzzle velocity minus initial, 2 km/s (kilometers/second) Thus t = seconds, a very short time. The acceleration needed to produce the 4 km/s muzzle velocity is: (2) a = dv / dt a = projectile acceleration (kilometers/second^2) dv = change in velocity, 4 km/s (kilometers.sec) dt = time spent under acceleration, s (seconds) Thus the acceleration is 1333 km/s^2. Talk about whiplash. The force on the projectile and thus on the rails as well is determined by the acceleration and mass of projectile:
46 The Big Bad Rail Gun (3) F = maF = force on projectile and rails (Newtons) m = mass of projectile, 2 kg (kilograms) a = projectile acceleration, m/s^2 (meters/second^2) Drum roll please...the force is N, or lbf. To get an idea of how much force this is, lets assume we wanted to keep the stress in the rail beam under 50 ksi (345 MPa): (4) A = F / tau A = area of beam in tension (square inches) F = force on projectile and rails, lbf (pound force) tau = stress on beam in tension, 50 ksi (thousand pound force per square inch) The cross-sectional area of beam required to meet this spec would be 11 in^2. That is a lot of metal. The above was for a purely tensile load, but the geometry of the actual situation is more complicated. The portion of the rail beam nearest the projectile will be in compression, with the farthest portion of the beam cross section in tension. The ratio of these stresses will depend on the cross sectional geometry of the rail and thus the position of the neutral axis. The neutral axis is the line parallel to the rail and the direction of projectile motion where there is no stress, compression and tension are in balance.
47 The Big Bad Rail Gun What kind of slugs (projectiles) can a rail gun fire? A rail gun can fire virtually any type of projectile, provided that a least some portion of it conducts electricity and makes contact with the rails. Home built systems often use quarters, washers, or ball bearings as projectiles. Mostly, these do not work, and just end up welding to the rails. Welding is the most common problem in home rail gun design, and is difficult to overcome using traditional methods. Some guns have used graphite projectiles in an effort to eliminate welding, provide lubrication, and increase rail life time. While solving the welding issues of more conventional types, these projectiles have several disadvantages: they are low density, and provide increased electrical resistance. How do I keep the projectile from welding to the rails? Welding is strictly a problem of current density and heat. Current density refers to the amount of current flowing through a particular portion of the circuit, and generally reaches its highest levels at the interface between the slug and the rails. The contact area between the slug and the rails must be as large as possible to keep the current density from welding the two together before the projectile can begin moving. In an advanced high power design, the projectile will ionize (can you say plasma?) due to the heat generated by the extremely high currents involved. The major difficulty in getting optimum rail to slug contact is the balancing act of projectile to rail surface are contact, aerodynamic stability of projectile, heat dissipation, and friction with rails. One solution to these issues, which results in exceptional projectile performance is the sabot round.
48 How does the sabot rail gun slug work?The Big Bad Rail Gun How does the sabot rail gun slug work? The main interest in creating a weapon that fires a projectile is transferring muzzle energy to the target through momentum (velocity and mass). At the same time aerodynamic stability and drag and thus weapon accuracy and range are of concern. The sabot round uses a disposable case (sabot) to carry the projectile clear of the weapon barrel. In the rail gun case, the sabot allows the best of both worlds, allowing the sabot to be designed for optimum current carrying capacity and heat dissipation while the projectile itself may be designed for stability and drag. Recent efforts use an aluminum sabot cradle that carries a tungsten, fin-stabilized round down the rails and off into and through the target. The cradle must carry all or most of the current provided by the power supply and thus contact area is very important. In these designs, the power is provided by a compulsator and often results in the ionization and vaporization of portions of the sabot cradle. Tungsten makes an ideal metal for the projectile due: it's high density (19.3 g/cc) , hardness (31 Rockwell C), and extremely high melting point (3370 C). See figure 1 for a sample rail gun sabot design.
49 Figure 1 Sample Rail Gun SabotThe Big Bad Rail Gun Figure 1 Sample Rail Gun Sabot **Note, I have been informed that the use of multiple fins in the sabot casing has been found to result in non-optimal current distribution and eddying. Current designs utilize a single contact sabot cradle. (Thanks Ben!) --- (Who is Ben????)
50 The Big Bad Rail Gun How do you fire a rail gun?Firing a compulsator-powered rail gun is a delicate undertaking. The basic idea is to tap the power signal at the desired voltage. This can be performed in several ways, with the simplest being using a shaft encoder to sense the position of the rotor in the stator and extrapolate this to a voltage. In a properly designed compulsator/rail gun design, you would fire at or near peak signal voltage at all times. However, if mission parameters call for projectiles of various mass, g sensitivity, or melting points the ability to vary the output voltage by tapping the signal at various points in the waveform will be desirable. As with a simple generator, the output across the compulsator terminals will be a voltage signal that varies in frequency according to the number of stator poles and shaft RPM. The more poles the compulsator has, the slower the shaft must spin for the output signal to reach a certain frequency. The rail gun is fired when a switching system closes the circuit between the compulsator output, the rails, and the projectile. The rail gun will continue to fire as long as the switch remains closed, the compulsator is spinning and powered, and the projectile is completing the circuit between the rails. In practice, the compulsator output signal frequency will have a much longer period than the time the projectile is in between the rails. With a 4km/s muzzle velocity and 10m rails, the projectile is in contact with the rails for seconds. If your compulsator has 8 poles and is spinning at 900 RPM, the output signal frequency will be 60 Hz and a period of seconds. In general a compulsator with a greater number of poles is desired in order to reduce the required rotational speed of the rotor assy. This is advantageous because of the size and mass required of the compulsator to withstand the forces and energy dissipation required of such a high power device. Thus the higher the required rotational speed the more critical rotor balancing and bearing forces become.
51 The Big Bad Rail Gun How can I switch the Rail Gun current?Closing the circuit between the compulsator and the rail gun itself is a design challenge unto itself. The "switch" must withstand all of the current that goes through the rails and projectile, which as we saw earlier can be very substantial. Basically, the switching mechanism MUST NOT weld itself closed. If it does, the pulses of current will continue to be sent through the circuit, possibly overheating your rails, thus warping, delaminating, melting, or destroying them. If the projectile has cleared the rails, this is of course of no consequence. However, as you will discover, working the bugs out of any system, especially a mega-joule rail gun, is a difficult task and you don't want your entire project to turn to melted slag if your projectile gets stuck in the rails. This said, you should design each separate component of your rail gun to be self-resetting or self-arresting. This means that the switching mechanism should not rely on the projectile to clear the rails to turn off the gun, the compulsator should not continue to spin or discharge unchecked, etc. The switching system must also introduce the minimum of electrical resistance to the completed circuit. Since the compulsator is a relatively low-voltage device (<1kV), all effort must be spent on keeping the resistance of the total circuit as close to zero as possible. In addition, the higher the resistance of the switch, the more heat it will generate and thus the faster it will destroy itself. As you may imagine this is a tenuous balancing act.
52 The Big Bad Rail Gun The best solution is good old solid state electronics. The SCR (silicon controlled rectifiers) is basically a diode that can be turned on and off. Designed for power systems, SCRs are available in ratings that will handle rail gun sized current pulses for short durations. By using a properly sized SCR or array of SCRs the compulsator power signal may be tapped using a a fixed or adjustable voltage sensing circuit that actuates the SCR at the desired point in the output wave. Proper heat sinking of the SCRs will determine their service life and rate of fire. As you may imagine, the SCR will introduce a voltage drop like any semi-conductor device. Another consideration of the SCR, is the turn off time. Once an SCR is triggered "on" it does not stop conducting until the voltage signal across it drops below a threshold level. In an AC situation like our compulsator, this is where the voltage signal crosses zero. So the longest pulse an SCR can pass is one half waveform, roughly from zero to peak and back to zero. Therefore, if the projectile has not cleared the rails, you must wait a half cycle to pulse again, or use two SCRs in a triac type formation to pass the last half of the wave, zero to negative peak to zero. This will result in a current through the rails of opposite direction as the previous pulse, and you will have to "destroy" the field established by the previous pulse before building the new opposite sign field. This will introduce phase lag, increased inductive impedance, and other effects that may be less than optimal. So, the point I am trying to make is, your compulsator and switching system should produce a pulse of sufficient duration and power to eject the projectile just as or just after the SCR shuts off. This balancing can be performed by rail length, peak voltage, pulse shaping, output frequency, etc.
53 The Big Bad Rail Gun How can I inject or load the projectile?As mentioned in other areas of the site, projectile/rail welding is a common problem in rail guns. In order to avoid this, many designs call for a mechanical injection system to give the projectile forward motion before it comes in contact with the rails and currents. As you know, kinetic friction is less than static friction. (think about four wheel drifting in your favorite automobile...) One important consideration when designing an injection system is of course timing. Making sure that the projectile has entered the rails while the compulsator is tapped at the desired voltage and so forth is very important, considering that the compulsator may be tapped for a duration of less than 10 msec. It is for this reason that highest power rail guns do not use an injection system, but rather a simple breech loading procedure.
54 The Big Bad Rail Gun If an injection system is desired, there are a number of ways to "squirt" the projectile into the rail assy. You can use spring-loaded, centrifugal, compressed air, or any other method you can think of. The best method is most likely compressed air, which can provide a large force and thus higher injection speeds. With a projectile cross section of 2 square inches and 50 PSI air, you could use PVC as your material and generate 100 pounds of force on the projectile. With a 1 pound projectile, this would generate an acceleration of roughly 3 g's. For timing, you could use an optical gate to switch you SCRs as the projectile crosses the breech of the rails themselves. This would allow fairly precise timing, as most SCRs have a fairly rapid response time (on the order of a dozen or so msec). By the time the compulsator is tapped the projectile would have traveled only a few inches.
55 ELEC 3105 Basic EM and Power EngineeringAnother Big Bad Rail Gun
56 The Big Bad Rail Gun Fatro and Jengel's Rail GunYour one stop, step by step guide to home rail gun technology Updated 9/17/99 Here lies the musings and possible truths of several engineering students on the subject of rail gun design. As of yet, all information is theoretical, which means we think it would work but are too lazy and poor to do anything about it. This page is provided for entertainment and informative purposes only and any connection to reality is sketchy at best. First off, for a rail gun to work you need a roughly uniform parallel magnetic field encompassing the rails and it needs to be perpendicular to the desired plane of motion. For example, if you want the projectile to go parallel to the plane of the earth's surface then you want your field to be oriented either pointing straight up or straight down. (See diagram)
57 The Big Bad Rail Gun So now we know what we need, but how do you generate a roughly uniform parallel magnetic field? The answer is using a solenoid. A solenoid is a coil of wire with N turns per inch and L inches long. The magnetic field created when a current is run through the coil is roughly parallel in the center of the coil (L/2). The strength of the field in an ideal solenoid is given by: B=Nui (1) Where B is the magnetic field, N is the turns per unit length, u is the permeability constant (u=1.26x10^-6 H/m), and i is the current through the wire. This equation suggests that the strength of the field in an ideal solenoid does not depend on the diameter. This equation holds for a real solenoid of finite length best at the center of the solenoid, at L/2, half the length. So we have a field, but what do we do with it? Since we need the rails to be perpendicular to the field, we'll have to run them in the coil, along its diameter. Thus, we'll need an aperture for the projectile to escape from. The rails should be as close to parallel as possible, and the projectile needs to be an electrically conductive object which spans the two rails, making solid electrical contact with each. The force applied to the projectile depends on three things, the strength of the magnetic field, the current through the rails and projectile, and the width between the rails. F=iLxB (eq 2) Equation 2 shows that the force is given by the cross product of the current times length with the B field. Thus, we see that we get maximum force when the current, and thus the L is perpendicular to the B (magnetic) field. See Diagram.
58 The Big Bad Rail Gun What now? We know that with a field and a current we can apply a force to an object, thus accelerating it, but how do we use this to build an actual rail gun? Our idea for a design was to use a ball bearing as a projectile, thus allowing less friction with the rails, a fairly aerodynamic shape and most of all ball bearings are cheap and abundant. For rails, we decided that angled steel would be best. The right angles can be used to confine the ball to one dimension of freedom and offer enhanced rigidity to resist the forces between the rails. This turned out to be a good idea in theory, but the area of contact between the ball and rails is insufficient, resulting in high current density and immediate welding. A traditional flat washer would serve better, as the area of contact limits the current that can pass through the projectile before welding occurs. Picking a gauge of wire for the solenoid is also important. You need to find the right balance between current capacity and turns per inch. Recall that the strength of the field generated relies both on number of turns per unit length and the current through each turn. Stacking layers of turns can be used to make up for a lower current, but each layer has a decreasing effect on the internal B field. Look for enameled winding wire, it has a thin painted on coating so that you won't get a short but it's thin so it won't limit the amount of turns too much. Wire like this (24 AWG) costs around 4 bucks a pound with about 1000 feet per pound.
59 The Big Bad Rail Gun Another problem with our design is that unlike other rail gun designs where the only circuitry is a bank of capacitors that discharge into the rails, our design builds a magnetic field in the coil before sending a current into the rails. In this way we can increase our field B, achieving a greater force without a greater current. However, it takes a discrete amount of time for the field to build in the coil, which is really an inductor. Thus as the field builds, the voltage across the coil (inductor) drops to zero as the derivative of the current through the coil drops to zero. Once the steady state value, or max current is reached, the voltage across the inductor is zero and current flows freely through the inductor. The result is that if our current in is a sinusoid cos(x) then the voltage signal would be cos(x-pi/2) = -sin(x) or lagging by a phase of pi/2. Our current in will most likely be provided by a car battery which is a 12V high amperage current source. As a result our input signal will be a unit step input with a finite rise time the voltage will lag here as well, being highest initially and then falling off as the current stabilizes. When properly designed, our rail gun will have a maximum allowable current right at the maximum output of our battery(ies), at this current the wire in the coils would melt, but we will design a cutoff circuit that will turn off the current to the inductor right before this point. This can be accomplished using a transistor cutoff circuit that senses the voltage across the rails and triggers a high current relay to turn off the voltage to the coil as this reaches a predetermined level. Another consideration is the fact that as the projectile moves down the rails, the magnetic flux will be increasing. As a result an induced voltage will oppose the voltage applied to the rails until the voltage across the rails is zero. This limits our rail length and thus the time that a force is applied to the projectile, limited by the strength of the magnetic field and the voltage applied across the rails.
60 The Big Bad Rail Gun Magnetic flux ø = BA for a uniform field over an area A Voltage induced= V(ind) = dø/dt = BdA/dt in our case since the field B is constant while the area is increasing. dA/dt = Ldx/dt where L is the width of the rails and thus the width of the projectile and dx/dt is the linear velocity of the projectile. As a result we have a terminal velocity for our rail gun. Max Velocity=v(max)= V(applied)/BL Finding an acceptable balance of cost and velocity is the hardest part of home rail gun design. Using two car batteries in series you can double the applied voltage and theoretically double the terminal velocity. In practice (mathematically) you will not reach the maximum velocity unless your rails are very long. Thus using 24V would increase the force on the projectile by increasing current, but be careful that you are not extending the draw beyond the capabilities of your battery. Most lead acid car batteries can put out 100 amps without breaking a sweat, so don't worry too much. Measure the DC resistance of your completed circuit before powering up to get a rough estimate of the current draw. Some other ideas when designing and building a rail gun are to limit welding, focus on increasing the magnetic field along the rails, and to increase the rail length to take full advantage of the forces you are generating. Several ideas that I have for this are as follows.
61 The Big Bad Rail Gun First, when a ferromagnetic material is placed in a magnetic field, the field permeates the material and locally aligns the magnetic orientation of the normally randomly oriented magnetic zones (due to grain structure in metals) along the direction of the applied field. Internally, this results in huge increases in magnetic field strength. This phenomenon is taken advantage of in electric motors and transformers. In addition, the internal alignment tends to focus the magnetic field lines in the vicinity of the ferromagnetic object, entering in the top (North) pole and exiting the bottom (South) pole. See diagram.
62 The Big Bad Rail Gun By using this effect to our advantage, we can concentrate the field generated by our solenoid into the area where the rails are, thus increasing the magnetic flux in the area of the rails. In this way the field strength is increased and thus the force generated on the projectiles. Proper geometry of the pole pieces placed within the coil can possibly concentrate virtually the entire field into the area of the rails, much like a primitive AC motor. This is of course only an idea. See below for more details.
63 The Big Bad Rail Gun In order to lengthen the rails, and thus the time the force is applied to the projectile and ultimately increasing muzzle velocity, we can squish the solenoid or even make it rectangular. By making the solenoid long and rectangular, we allow ease of finding pole pieces to capitalize on the effect mentioned above as well as limiting the space our rail gun will occupy and the wire needed. The drawbacks to changing the geometry is making calculation of the field within the solenoid very difficult. It may be adequate to assume uniform dispersion, but due to geometric effects, the field will vary along the length of the solenoid, with areas of unknown or hard to predict strength at each end. This is problematic in that a strong field is needed when the projectile is beginning to be accelerated in order to limit welding tendencies. One way to overcome this, and most likely a good idea in all rail guns is to have a mechanical injection system. A mechanical injection system imparts the projectile with an initial velocity as it enters the coil and makes contact with the rails. Problems with building a system such as this are the mechanical complexity, timing the electrical circuits to synch with the injection, and ensuring good electrical contact of the projectile with the rails. One idea that I had for bypassing most of these caveats is to use a surplus pinball launcher, the thing that you pull back to shoot the ball. Using this, and a simple switch or two, you could jump start the projectile and start the current flowing as the plunger pushes the projectile into the rails. Another alternative is to use a smaller solenoid actuator with a iron rod that moves in and out according to voltage applied to smack the projectile. Use of a swinging arm like a clay pigeon launcher could also be used but would be more complicated. Use your imagination.
64 The Big Bad Rail Gun Further design enhancements that would help rail gun operation are using conductive greases to initially maintain forward motion since kinetic friction is less than static friction (once something starts to slip, it tends to continue slipping due to this effect). However, once the grease has heated up enough it will no doubt break down and possibly necessitate cleaning before firing again. The use of a plastic rail follower to maintain constant pressure on the projectile could help keep current densities below the welding point by maintaining good surface area contact. Conductive powder based lubricants like graphite could also be used, but due to the decreased conductivity could limit current. Ramping the current through the rails up after projectile injection and rail contact has been made could also help alleviate welding by allowing the projectile to build speed before applying the full current.