General Railgun Function

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Buck Rodgers
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1 Geneal ailgun Function An electomagnetic ail gun uses a lage Loentz foce to fie a pojectile. The classic configuation uses two conducting ails with amatue that fits between and closes the cicuit between the two ails. A lage cuent is sent though the ails fom one side ceating a magnetic field that is pependicula to the cuent in the amatue and foces the amatue to move down the ails away fom whicheve end the cuent is being applied. Since the foce pushing apat the ails is lage as well, they must be built so as to withstand the foce developed duing fiing. In ode to enhance the magnetic field developed acoss the amatue seveal additions can be made to augment the classic configuation. A coe with high magnetic pemeability may be used to incease the magnetic field density, although this may not be helpful if the ailgun opeates well past the satuation point of the mateial. A stong pemanent magnet cicuit may be used to povide a magnetic field sepaate fom that povided by the ailgun, again that may not help much if the ailgun opeates with a highe magnetic field than the pemanent magnets. Additional ails that do not make contact with the amatue may be placed nea the main ails in ode to incease the magnetic field density. Any of these options could also be implemented with a completely sepaate synchonized cicuit. The amatue can also be the pojectile, o the amatue can act only as a sabot o pushe fo a pojectile. The amatue can be solid o plasma. The plasma being supplied by a thin conductive metal that is heated apidly by the cuent though it. If a plasma amatue is used the ails, and the stuctue holding the ails togethe must povide a completely sealed bael that closely fits the pojectile so that the plasma does not blow past the pojectile. The powe supply to fie ailgun must povide a vey lage cuent of shot duation. It must also opeate at a high enough voltage to dive the equied cuent and to squelch any back emf fom the amatue. DC supplies such as lead-acid supplies can delive seveal thousand Ampees fo shot duation, but ae not pactical fo a lage weapon since a lage amount ae needed to povide the equisite voltage and cuent. capacitos and compulsatos can stoe vey lage amounts of enegy and ae capable of deliveing hundeds of kilo-ampees. Capacitos stoe enegy via an electic field; compulsatos stoe enegy mechanically in a flywheel. Compulsato stands fo Compensated Pulsed Altenato; a compulsato uses a vey low inductance geneato to allow fo apid cuent ise and a high enegy density flywheel to stoe enegy. A compulsato can stoe enough enegy to fie a ailgun seveal consecutive times whee a capacito bank usually uses all of its enegy on each shot and needs to be echaged afte each shot. Compulsatos geneally can stoe moe enegy pe unit volume than capacitos.

2 Fo any ailgun the cuents equied will place lage amounts of mechanical stess on the cuent caying pats. The ails cuent caying bas and connectos must be stiff and fastened into place. If the ailgun uses a plasma amatue the amatue/pojectile will need to fit tightly into the bael and the bael will need to be sealed to keep the plasma fom escaping. Fo a solid amatue the suface aea in contact with the ails will need to be maximized and good contact maintained, this is necessay to educe acing, spot welding and to allow fo high cuent flow. The amatue ail inteface should be designed to minimize gouging, and if any gouging is to occu it is clealy moe desiable to have the damage on the amatue instead of the ails. So the ails should be as had as feasible and the amatue as soft as possible. Many mateials and constuction techniques have been tied to make long lasting ails. This is still an aea of significant eseach. When the ailgun is fied the amatue/pojectile should be injected at a high velocity to ovecome that static fiction of the bael and to pevent spot welding. A fast injection also will spead out the heat geneated acoss a geate aea again helping to polong ail life. Unless the magnetic field is supplied extenally the amatue should have a sufficient length of cuent caying ail behind befoe it contacts the ails it in ode to allow stong a magnetic field to be ceated. Even so it may be desiable to augment the magnetic field whee the amatue fist makes contact since the cuent will not immediately begin to acceleate the amatue. ailgun Foce and Magnetic Field Analysis The geneal Loentz foce law descibes the foce on a moving chage in magnetic and electic fields. (1) F = q E q v B [1 pg 8] whee is the foce acting on q [N/C] E is the electic field intensity [N/m] B is the magnetic field intensity [T] v is the chage velocity [m/s] q is the chage [C] In a ail gun the moving chage consist of a lage electic cuent in a conducting amatue, the foce to acceleate the pojectile is given by the second tem in (1). Since cuent is defined as a movement of chage pe second, a cuent ove a length l is equivalent to chage movement at a given velocity.

3 m s C s () q[ C] v = i l[ m] (3) F = q v B = i l B Expessing the coss poduct in (3) using the angle theta between the cuent and magnetic field gives () () F = i l B sin( θ ) If the magnetic field is assumed to be pependicula to the cuent then () educes to (5) F = i l B Whee B is the magnitude of the magnetic flux density. A ail gun can povide the cuent and magnetic flux using a conceptually simple cicuit. FIGUE 1. The basic ailgun concept, two paallel ails and a sliding amatue. The cuent in the ails and amatue ceates the magnetic field indicated and a the desied foce on the Pojectile/Amatue. Fo a given cuent the foce along the amatue vaies along its length, and the magnetic field vaies accoding to the length of the ails. The φ diected magnetic field of an infinitely long wie is

4 (6) B a = π i φ whee is the pemeability constant is the adial distance fom the wie The contibution fom a single vey long, thin ail would be ½ that of an infinitely long wie (7) B a = φ i π The ails that ceate the magnetic field may not be well appoximated by assuming they ae infinitely long. An expession fo the magnetic field at a adial distance fom the end of a wie can be found using the Biot-Savat law. J a 1, 3 dv V π (8) B( u u, u ) = ' [1, pg 75] whee J is the cuent density [A/m ] is the distance to the point of the contibuted magnetic field if it is assumed that the coss section of the wie is negligible compaed to the aea to be integated ove then Jdv' = i dl and (8) educes to (9) ( u1, u, u3 ) i dl a B = l π Fo the puposes of the following integation it assumed that the ails ae vey thin.the ails have length L, the magnetic field at a distance d fom the end of one ail is then (1) B ( P) = L i dl sin π ( θ ) the distance fom a segment of wie to the point P can be expessed as (11) = l d

5 the quantity sin ( θ ) can be expessed as (1) sin ( θ ) = d = l d d the integal then becomes (13) B ( P) i dl L l d i L = π d L d = l d π d When L becomes much geate than d the equation educes to (7). Since the ails of a pactical ailgun cannot be infinitely long and a signifacant magnetic field is desieable it is useful to know how long the ails must be to easonable appoximate infinite ails. Expessing L as a multiple of d, L = x d, setting d equal to 1 and dividing (13) by (7) gives a function descibing the faction of magnetic field compaed to a wie of infinite length fom the beginning of the ail. The faction of magnetic field poduced by finite length wies compaed to infinite legnth wies as a function of the atio x of wie length to the point whee the magnetic field is measued. (1) x 1 x The elation shows that when the wie thee times longe than the distance to the point at which a given magnetic esults, the field density is 95% of that of an ideal infinite wie. Equation (1) assumes a vey thin wie, this is not the case in a ail gun, the dimensions of the pojectile ae of the same ode of the ails caying the cuent. Howeve, this elation does give some idea of the length of ail necessay to set up a satisfactoy magnetic field.

6 FactionOfIdeal FIGUE. The faction of an ideal magnetic field ceated by finite ails in tems of the wie length in multiples of the distance to the point the magnetic field is calculated. atio Assuming that the ails ae constucted such that the magnetic field ceated closely appoximates that of thin wies that extend to infinity, equations (5) and (7) can be combined into an expession fo the foce on the pojectile in tems of cuent and length. The magnetic field contibution fom the fist wie being equivalent to equation (5) and given a ail sepaation of width w the expession of the magnetic field contibution fom the second wie is (15) B = a φ i π ( w ) The total magnetic field at a point along the amatue is then i i (16) B = a φ π ( w ) π ( ) Since the amatue is in diect contact with the ails the magnetic field fom the above equation would appoach infinity at the points whee the amatue contacts the ails. So despite the fact (16) assumes a thin long wie the adius of that wie must be taken into account to pevent spuious esults fo magnetic field stength at the ends of the amatue. Taking into account the adius of the ails gives

7 (17) = ) ( ) ( π π φ i w i a B Inseting (17) into (5), keeping in mind the cuent and magnetic field ae pependicula thought the amatue gives the diffeential foce (18) = = ) ( 1 ) ( 1 ) ( ) ( π π π w dl i i w i dl i df These equation show that the geatest foce on the amatue occus at the point whee the amatue is closest to (touching) the ails. As the adius becomes smalle with espect the ail sepaation the dispaity in foce fom the cente of the amatue compaed the foce at the ends of the amatue becomes lage. Since the smallest foce on the amatue occus at the cente of the amatue and lagest foce occus at eithe end setting the distance ho in (18) to and dividing by (18) with set to w/ will give the atio of foce on the end of the amatue to the foce on the middle of the amatue. (19) ( ) ( ) w w Setting w in equation (19) equal to 1 allows fo a gaph of the foce atio as a function of wie adius in multiples of w.

8 FIGUE 3. The foce dispaity between the ends and cente of an amatue as a function of the atio between the ail adius and distance between ails. Integating (18) ove the the gap between ails gives an expession fo the total foce on the amatue. () = w i 1 1 i w F d = ln π ( w ) ( ) π Equation () descibes the foce exeted on the pojectile a function of cuent, ail sepaation and ail adius. To get to this point I have assumed ound wielike ails of adius, and I have assumed that the distance fom the ails along ail sepaation w is lage compaed to the dimensions of the ail. The second assumption is clealy incoect fo points vey close to ails, and if the ails ae simila in size to the sepaation between them then the assumption is false at all points acoss the ail sepaation w. I have also assumed the ails ae long enough such that the magnetic field acoss the amatue is the same as if the ails stetched out fom the pojectile in infinite length. Figue shows that this is a good appoximation if the ails ae seveal times longe than the ail sepaation. Anothe assumption is that the pojectile is vey thin. Yet anothe is unifom cuent density. In ode to get a bette equation fo the foce exeted on the pojectile the Biot- Savat law, equation (8) would need to be applied pecisely to the ail foms. The aveage magnetic field acoss the amatue would equie integating the equation acoss the ail sepaation w. Since the Biot-Savat law is a volume integal, the total equation would be quaduple integal. Doing this would eliminate the assumptions of half-infinite ail length, ound ails, and the poo assumption that the points along w vey fa way fom the ail compaed to the dimensions of the ail. An analytical solution could be obtained fo diffeent types of ails, but a vey complicated shape could esult in an integal esults in a long ugly esult, o one that can only be solved numeically. Taking away the assumptions of unifom cuent density and a thin pojectile petty much foces one into finite element analysis and othe numeical techniques to analyze ailguns. Howeve, despite its limitations equation () is useful. It gives appoximate cuents needed to poduce a given foce. Moe impotantly it gives some pointes as to how to design a ailgun. It shows that the foce can be made much geate using magnetically pemeable mateials. In the ange of cuents and magnetic fields found in ailguns, µ is not constant. It deceases significantly at vey high fields, limiting the usefulness of pemeable mateial. Anothe limiting facto of using pemeable mateial is enegy loss due to magnetization, and eddy cuent losses. Equation () also shows that the ail sepaation w should be maximized, although this value is set by the choice of calibe. The final impotant obsevation fom the equation is that the ail adius should be minimized. Fo

9 ectangula ails this means they should not be vey tall o wide compaed to the bael. This makes sense, since the cuent obviously needs to be concentated nea the bael in ode to ceated the lagest magnetic field possible though the amatue. The conclusions egading ail sepaation and ail dimension have been confimed by eseaches using Finite-Element Methods []. Dividing the foce of equation () by the iw will give an expession fo the aveage magnetic field accoding to equation (5). (5) F = i l B eaanging (5) and eplacing l with w since w is the amatue length in ou equations gives F i w (1) B = = ln i w πw Equation () and (1) allow fo calculations to estimate the magnitude of cuents and magnetic fields needed to fie a pojectile fom a simple ailgun. Back EMF fom ailgun pojectile. The voltage of couse needs to be sufficient to dive the equied cuent though whateve esistance the ails, amatue and connections povide. Apat fom this the voltage must also be lage enough to ovecome the back EMF fom the pojectile. d () E dl = B ds l dt S When the ails ae elatively long the magnetic flux being ceated behind the pojectile is constant and ate of change of aea is the poduct of the amatue length and its velocity. The line integal on the left hand side of () can be assumed to include the amatue and diffeential aea behind it, and the ight hand side is simply the ate of flux change in the aea behind the pojectile. Equation () then becomes. (3) E = B v w whee B is the aveage magnetic flux density [T] v is the velocity of the amatue [m/s] w is the amatue width [m]

10 E is the back EMF developed acoss the amatue [V] The back EMF will oppose the established cuent, but given estimates of the expected magnetic field stength fom (), the expected velocity and the width of the amatue a voltage souce can be chosen so that the ailgun can fie at the desied velocity. Mechanical Enegy Stoage and Compulsatos. A compulsato stoes its enegy in a flywheel connected diectly to the altenato. Assuming the flywheel is a solid homogeneous cylinde of mateial, it otational inetia is 1 I = M The otational kinetic enegy is 1 K = Iω the mass of the cylinde is M = l π Combining these equations into an expession fo kinetic enegy gives 1 K = πl ω Expessing this as enegy stoed pe unit mass gives K = 1 ω This shows that a oto of high speed and lage adius is desiable in ode to stoe the lagest amount of enegy possible. Depending upon the mateial chosen thee is a limit as to how fast the oto can be spun. Each mateial has a maximum peipheal velocity at which the centifugal foces ovecome the tensile stength of the mateial. The peipheal velocity is v=. The maximum peipheal velocity is given by

11 V = max E tensile Since v=. The maximum velocity equation can be inseted into the enegy stoed pe unit mass to give a maximum enegy stoed pe unit volume. K 1 = ω = v 1 = 1 Etensile This shows that a high tensile stength to density atio is desiable in an enegy stoage oto mateial.

12 EFEENCES [1] Johnk, Cal T.A.. Engineeing Electomagnetic Fields and Waves. New Yok: Wiley, [] Keshtka, A.; Effect of ail Dimension on Cuent Distibution and Inductance Gadient, IEEE Tansactions on Magnetics, Vol 1, no 1, pp May 5.

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