Skin and heating effects of railgun current

zyxwvutsrqp zyxwvutsrq zyxwvutsrqp IEEE TRANSACTIONS ON MAGNETICS, VOL. 2 5 , NO. 1, JANUARY 1989 381 SKIN AND HEATING EFFECTS OF RAILGUN CURRENT J. C. Nearing and M. A. Huerta University of Miami Physics Department Coral Gables, FL 33124 Abstract Introduction In a railgun the current goes into one rail, passes through the armature, and returns via the other rail. The armature is immersed in the magnetic field produced by the rail currents and experiences a strong magnetic force that pushes the projectile in front of it. The distribution of current in the rails is one of the important factors in the operation of railguns because it determines the rail resistance and the joule heating losses. A great deal of work has been done to model this current distribution. Kerriskl>’ did a numerical calculation of the current distribution and of the accompanying temperature distribution, allowing the electrical conductivity to depend on temperature and the magnetic permeability to depend on magnetic field strength. He treated carefully the two dimensional variations in the rectangular cross section of the rails but neglected the variations in the direction along the rails. Marshall3 discussed qualitatively the problem of the variation along the rails. Long4 attempted a solution for a case with steady current and speed. Drake and Rathmann’ obtained infinite series solutions that described the variation of the skin depth along the rail due to the motion of the armature. The above papers do not describe the way the current turns the corner as it passses from the rail into the armature. Young and Hughes‘ have published an interesting approximate solution to this problem in the special case of constant current and armature speed. In a railgun, the part of the rails ahead of the armature is in a region where there is almost no magnetic field. As the armature sweeps along the rails, the rails are exposed to the strong field behind the armature. Due to the rapid motion of the armature along the rails, the current and magnetic field do not have time to diffuse completely into the rails and are concentrated in a skin layer near the rail surfaces. We seek to describe analytically how the magnetic field and current * zyxw diffuse into the rails and what the temperature distribution caused by the Joule heating is. The important skin depth that arises can be written as 6 = where t is the time since the magnetic field started to diffuse into the conductor of conductivity U , and magnetic permeability po. There are several related problems presented by Knoepfel’ with known analytic solutions. They involve the calculation of the magnetic field and current inside a semi-infinite space bounded by 7. plane, or inside a cylinder, when the magnetic field outside is uniform in space and has a known time dependence. zyxwvuts zyxwvutsrqpon zyxwvu zyxwvuts zy I We present a calculation in a simplified geometry for the current distribution in the rails, taking into account the motion of the armature and the time variation of the current. Closed form, asymptotic, results for the current density are obtained for arbitrary time dependent currents and velocities, in the limit in which the length scale k c 1 is small, where ko = (pocrv). U is the electrical conductivity of the rails, and v is the speed of the armature. Because of eddy current effects the rail current may reverse in portions of the rails when the total current decreases. The current is used as a source of Joule heating to find the temperature distribution in the rails. The heat diffusivity is negligible and we are able to give numerical results concerning melting.* d-, In order to elucidate the effects of the motion of the armature, our analytic solution takes into account the variations along the rail direction, z as shown in Fig. 1. The rails are t‘ , upperrail -xo(0 armature current breech lower rail 3 2 muzzle W I 6 Figure 1. Schematic of the rail geometry taken as infinitely thick because their thickness is much greater than the skin depth. We also take into account variations in the current density in the direction into the rails, y. This will enable us to describe skin depth effects on the inside surface of the rails. We simplify the geometry by neglecting variations in the z direction, in effect making the problem that of two rails of infinite height. In a complete treatment of the problem J y ( t ) in the armature would be determined by the properties of the armature in conjunction with the properties of the rails. We model the armature by an assumed time dependent distribution of current density in an armature moving to the right along the rails with an arbitrary velocity vo(t) = d z o / d t . We obtain rigorous general results for the current distribution in the rails and find the temperature distribution for special cases. We get conditions for melting depending on the current, and much less strongly on the length of the arc. We also give an expression for the breech voltage in terms of the current, and discuss the circumstances when it reduces to the simple expressions that are typically used in railgun circuit models. One of the most interesting results is the reversal of current density in the rails, and possibly in the armature, whenever the total rail current decreases. Current Calculation This work supported in part by the Air Force Office of Scientific Research under grant number 84-0116. In the coordinate system shown in Fig. 1, the upper rail oO18-9464/89/01OO-038 l%Ol.oOO1989 IEEE zyxwvutsrqpo zyxwvutsrq zy zyxwvuts zyxwvutsrq zyxwvutsrqpo zyxwvutsr zyxw zyxwvutsr occupies the space y > 0. The numbers 1-6 are used later to identify important voltages. The lower rail occupies the space y < --w and carries a current that is a mirror image of the current in the upper rail. The armature between the rails occupies the space -tu < y < 0 and has a current density J,(z,t) Jy(z,t ) = - I ' ( t ) f ( z - zo(t)) where I' is the current per rail height, f represents the spatial distribution of the current in the armature, and z o ( t ) is the position of the center of the armature. f is normalized to j f ( z ) d z = 1. Everything is independent of z, and only the z, y, and t dependencies are to be determined. To analyze the fields resulting from this source, we write Jy as a Fourier integral zyx The coefficient for F ( y ) comes from the continuity of B, at y = 0. Since the fields and currents must die off as y becomes large and positive, the sign of the square root is %(a) < 0. In order t o obtain the value of the magnetic field for y > 0, we integrate over k and w in Eq. (1). The value of B for y > 0 will then be B,(z,y,t) = -ipo J d t ' J 2n J dw 2n 1 k (5) .~ l ( t l ) ~ ( k ) ~ " y + ' ( k " ---i(kzo(t')-wt') wt) Thew integral can be done with no further assumptions on the time dependence of 1',or the shape f of the armature's current distribution. The integral e"Y-'w(t-t')dw/2n is zero for t < t'. The integrand has a branch point at w = - i k 2 / p o a , so when t > t', we deform the contour around this branch toward -im. Change variables t o s = iw - k 2 / p o u and observe that the imaginary part of the integral vanishes. Using integration by parts we obtain zyxwvutsr In Maxwell's equations we neglect the displacement current compared to the conduction current because all the time constants considered are much larger than €,/U. The resulting equations are VxB=poJ aB VXE=--. and at In the domain y < 0, a vacuum, the magnetic field due t o a current density Jy = ei(kz-wt) is B, = 'PO ei(kz--wt) (Y < 0). Taking the full Jy of Eq. (1) into account we find the magnetic field for y < 0 to be The integral of Eq. (6) is a general expression for the magnetic field in this geometry. (The behavior of this expression as y + 0 needs t o be treated carefully.) In order to make further progress on this integral, it is necessary to assume a shape for the current distribution in the armature. We have considered several forms for f(z). One of the simplest forms t o take that yields interesting results is 1 f(z) = - ( - L < z < L). 2L With this form, the k integral of Eq. (6) becomes 00 B(Z,Y,t) = - P o I ' ( t ) i dz'f(+') (Y < 01, (2) where 6 = z - zo(t). The absolute value 161 is the distance along the rail from the point with coordinate z t o the center of the armature. In arriving at ( 2 ) we have used the integral 4 J'" 2n = pm 1 z<o (0 z>o where the path of integration goes above the pole at k = 0. The magnetic field of Eq. ( 2 ) is constant behind the armature, where t < 0, and zero ahead of it, where 6 > 0, as one would expect in this geometry. In the domain y > 0, the interior of the upper rail, the electric field is eliminated using the conductivity equation, J = oE. Eliminating E and J from Maxwell's equations gives the diffusion equation for the magnetic field dB V2B = poa- at . (3) The vector B has only a t-component. A single Fourier component is written B, = F ( y ) exp [ i ( k z - ut)] (y > 0). When this is substituted in Eq. (3), the result for F ( y ) ~ ( y =) iP0 emy, a= d-. (4) J dk 1 sin k L e i k z - P k ~ 2n k k L The contour goes from -m to +CO, passing above the singularity at k = 0. Bring the contour down to the real-k axis, and we get a principal value integral plus a semi-circular contour just over the pole at zero. Only the cosine part of exp(ikz) contributes t o the latter, and only the sine part to the former. We get 00 sin kL sin k z d k T T -i+il We combine the two sines into the difference of cosines and use a tabulated' integral t o obtain zyx zyxwvutsrqpon zyxwvutsrqp zyxwvutsrq 383 The above expression for the magnetic field allows an arbitrary thickness L for the armature, an arbitrary time dependence for the current I ( t ) ,and a general time dependence for the armature position zo(t). The curl of this expression for B, will give the current density. Rather than writing down the complicated expressions that result in the general case, we will specialize to two cases of interest. In the first case we consider a thin armature ( L + 0) with a general time dependent current and velocity. The thin armature case simplifies the derivation of an approximate current-voltage relation for the railgun. It also allows a closed form analytical expression for the temperature rise caused by very concentrated currents. In the second case we will consider a thick armature but with the current and the velocity kept constant in time. This case is of interest because we are able to calculate the temperature rise of the rails for realistically large arcs. Time deDendent results for a thin armature with imaginary argument. At a distance from the current source large compared to k;', the current density of Eqs. ( 1 2 ) and (13) has the asymptotic form' From here one can show that the shape of these flow lines is parabolic at large distances from the origin. In Eq. (14) the space in front of the armature has E > 0 and T > 0 so the exponential is damped very quickly (for large lco) giving negligible current ahead of the armature as expected. Behind the armature, 6 < 0, there is damping for y > 0, but along the rail surface the exponent is zero. zyxw zyx The current of Eqs. (10) and (11) can be evaluated asymptotically for a general armature current and velocity. The integrals are approximated using the method of steepest descent; we have done this calculation. It is straightforward but somewhat long. In the region where zo(t) - z >> y > 0 there is a much simpler way to obtain the current. For large U the argument of the error function in Eq. (9) varies rapidly. We make the approximation that the error function is a step function, with 1 - erf(z) w 2 for z < 0, and 1 - erf(z) x 0 for z > 0. Then the magnetic field of Eq. (9), behind the armature where i$= z - z o ( t ) < 0, is approximately zyxwvutsrqpo zyxwv zyxwvutsrqp zyxwvu A thin armature could represent current spots or filaments. In this section, then, I ' ( t ) could represent the current in a spot, and not the total current through the rail. Of course, due to the two dimensional nature of our model, the spot is really modeled as a sheet. The thin armature result is obtained by taking the limit as L 4 0 in Eq. (7) dse-"I'(t - up0y2/4s2), The curl of this expression for B, will give the current density J. The singularity as y -+ 0 is best treated by rewriting B, as with s, = y / 6 , and we have introduced a skin depth 6 defined by where t , (9) where t' = t - y 2 p o u / 4 s 2 . The form of (9) is well behaved as y + 0. The current is (15) < t is the time when the armature passes position z o ( t z )= 2. Eq. (15) has a pleasing form and is similar to some of the results in K n ~ e p f e l .We ~ obtain the currents from the curl of Eq. (15), z, defined by m J,(z,y,t) = - q y l d k k e - k a y Z I ' ( ~ ) e - U a , (11) where U = k [ z - ~ o ( T ) ]and , T = t special case is where the current is the velocity of the thin armature is The above integrals for the current closed form throughout the rails, - p o u / 4 k 2 . The simplest constant, I ' ( t ) = I ' , and also constant, zo(t) = vt. can then be evaluated in where v, is the velocity at time t,. We have written J , in Eq. (17) so that the first term contains I ' ( t ) . If that term is combined with the second term in the integral it can be seen that the first term would then contains I'(t,). We note that even if I ' ( t )remains positive, J , ( z , y , t ) may become negative because of the term with d I ' / d t , whenever the current decreases. Eqs. (17) and (18) are very accurate when we are just a few lengths (pOuv)-' away from the present position of the armature. This was verified by extensive comparisons with numerical evaluations of the exact expressions in Eqs. (IO) and (11). The length hi' is very small in typical rail launcher situations. For the case of copper at room temperature, and a zyxwvuts where ko = pouv is an inverse length that can be used to scale all of the variables, T = is the distance from the field point to the present position of the armature, 6 was defined below ( 2 ) , and the K's are Bessel functions of the second kind d m zyxwvut zyxwvutsrqpo zyxwvutsrq zyxwvutsrqpon zyxwvutsrqpon meters. speed of 1 km/sec, the length H i 1 = 1.25 x 0.12cm current flow 1 T t 1.75 cm Figure 2. Current in the rail A picture of the current flow for these parameters is shown in Fig. 2. The acceleration is 8 x lo7 m/s2. Here the lines are tangent to the current density vector J and the curves are drawn at equal increments of current; the last curve includes 95% of the total current. The current has passed its peak and is decreasing. Some reverse current flow is evident in the graph in regions where Jz(x,y, t ) has become negative. This should not be surprising. This reverse current is just an eddy current that is trying to keep the magnetic field in the rail from dropping in value as the total rail current decreases. The total horizontal scale is 1.75 cm; the total vertical scale is expanded to 0.12 cm. why the muzzle voltage in Eq. (20) is given by just the resistive drop in the armature. We use the path of integration 1-2-3-4 shown behind the armature in Fig. 1 t o find that the breech voltage v b is where xo(t) is the position of the armature at time t . Here we have used the fact that in this geometry B , is uniform behind the armature and that the integral 2 -+ 3 is the same as the integral 4 -+ 1. The breech voltage Vb(t) involves the integral of J, dong the rail's inner edge, y = 0. Equation (17) shows that Vb(t) does not depend only on the values of I' and dl'fdt at time t, but on their time history. The circuit equation will therefore be an integro-differential equation. If dI'/dt' in the integral is not too large, as is the case in most rail launchers, we can simplify the result by doing a Taylor expansion around the upper limit of integration and carrying out the t' integral. The result is zyxwvutsrqpon zyxwvutsrq Breech and Muzzle Voltages There are several circuit models7*' used to describe rail launchers as circuit elements; these require expressions for the voltages in terms of the currents. These expressions can be obtained by application of the theorem i / B .dS = dt / [E V dt + + -301( t - toy? dt + *. . as an expansion in powers of ( t - t o ) . Especially nearer the armature, the contributions of the higher derivatives of I' in Eq. (22) may be neglected compared to the first derivative. We can now use these results in writing the breech voltage of Eq. (21). We use the fact that behind the armature, between the rails, B ( x , 0, t ) = /.ioI(t)/h, where h is the rail height, and I(t) is the total rail current, I ( t ) = h I ' ( t ) ,t o write 1 zyxwvutsrqpo zyxwvuts - x (U x B) f ( ' 7 . B ) ~. d S , where the surface integral is taken over a surface moving with local velocity U. For the magnetic field B, using Maxwell's equations, we obtain d /B 2 J,(z,O,t) z - I ' ( t ) 6J;; 1 d21' - -(t - to)6 dt2 .dS = - f [E + U x B] . d r , (19) where the closed line integral is taken over the moving path. In a moving conductor Ohm's law is J = u(E U x B). For simplicity we will derive the circuit equations using the thin armature results. Integrating Eq. (19) around the path 3-4-5-6 shown ahead of the armature in Fig. 1, we find that the muzzle voltage V,,, is + V5 - Vs = V m ( t )= V a ( t )= / O --w Ja(t) -dyS4, Oa (20) where Ja and U , are the current and conductivity of the armature, w is the rail separation, and V , is the resistive voltage drop across the armature. We have used the fact that B is zero ahead of the armature in this simple geometry and that the current is extremely small ahead of the armature. This is The first term in vb(t) allows us in this simple geometry to calculate the usual rail inductance W Lo(t) z POhZO(t) (24) proportional to the distance z that the armature has traveled. The second term is the rail resistance term. The skin depth that enters there involves the time t - t o , which is the time since the armature passed position I. The term Va(t)is the resistive drop in the armature. The last derivative term in Eq. (23) is a skin inductance term that is small compared to the main inductance in typical railgun situations. The ellipsis indicate the presence of second and higher derivatives from Eq. (22). These would be negligible only for slowly varying currents. The effective resistance of each rail can be read from Eqs. (16) and (23). We use the definitions oft, and 6 to write the rail resistance at time t as (25) zyx zyxwvutsrqponm zyxwvutsrqponm 385 zyxwvutsrqpo zyx zyxwvut zyxwvutsr zyxwvutsrqpo where v o ( t ) = d z o ( t ) / d t . R ( t ) can be expressed in terms of the instantaneous armature position x only if x is known as a function o f t . We note that an elementary derivation of R ( t ) would miss the f i factor. For example if the armature speed is constant we get coordinate z to the center of the armature. The +-component of the current density at the y = 0 rail surface is given by I' J,(z,O,t) = 27rL [ M ( k o ( ( L ) ) - M ( k o ( [ - L ) ) ] + where where we used z = vot. This implies that there is no fixed resistance per length, but rather that the resistance varies more slowly with length than linearly. The case of constant acceleration, z = at2/2 gives Temperature Distribution The local heating of the rail is proportional to the square of the current density. Solve the heat diffusion equation dT 52 - K V ~ T= -, dt U pc- neglecting the latent heat of melting, where c is the specific heat per mass, p is the mass density, K is the heat conductivity, and T is the temperature. The general solution of this equation in our geometry can be given in terms of Green's functions. We only need however, the case of small dimensionless diffusion coefficient, D = Kpou/pc <( l . (D = 0.0053 for copper at room temperature.) In this case one can verify that the the heat diffusion is negligible eveywhere for an armature of nonzero thickness. This approximation, for the case of zero armature thickness, exhibits a spurious logarithmic singularity at the surface as will be discussed below. With this approximation, then, during the short time that the current flows, the heat generated at a point in the rails, 1dt J 2 / u , does not have time to diffuse away but rather it stays where it is produced. The temperature rise at a point is then This formula implies that the rate of heating is greater at points near the arc, but it also says that the higher temperatures are reached at points farther behind the arc (closer to the breech). This happens because the heat diffusion in this short time is so small, and because points near the breech are subjected to the current for a longer time. Of course radiation cooling can affect this. First we will calculate the temperature at the rail surface for the case of a thick armature of length 2L. The special case where both the rail current and the velocity are constant in time allows us to express the current at the rail surface in closed form. Due to the lack of heat diffusion this will be the most important part of the current needed to understand the ohmic heating and melting of the surface. The y-component of the surface current density is zero except in the armature region from f = -L to f = +L, where it has the value I'/2L. Recall that I f 1 is the distance along the rail from the point with (28) M ( x ) = [(x + 1)K0(\zl/2)- lz1K1(1x1/2)]e-2/2 The temperature at the surface can be found in the approximation of Eq. (27) using the current of Eq. (28). The integral is readily evaluated numerically. The temperature is poI'2/pc times a dimensionless factor depending on koL and ko[. For k0L = 25 and k o f = -50000, the factor is 3.04; for k0L = 25 and kof = -5000 the factor is 2.31. For koL = 2000 and the same f ' s , the factors are 1.63 and 0.89 respectively. Again we take copper at room temperature and a speed of 1 km/sec, where the scaling factor ko = 8 x l o 4 m-', so a kof of 50000 becomes 0.63 m; k0L = 25 gives an armature width 2L of 0.6 mm; k0L = 2000 corresponds to the width 2 L of 5 cm. Using the room temperature values of p and c , and a typical value of the current density 3 x l o 7 Amplmeter, the factor 3.04 gives a temperature rise of 985 C. The factor 1.63 gives 528 C. The melting point of copper is 1083 C so these examples do not lead to melting. The temperature rise is far less sensitive to the length of the arc than it is to the overall current per height of the rail. This is essentially because the penetration of the current into the rail is small for a long time; so, even if the length of the arc is long, all of the current will eventually have to pass near any given part of the surface. That the current does gradually move into the rail is reflected in the weak sensitivity of the temperature to the arc length. A factor of 80 in k0L causes a factor of less than 2 in the final temperature. The weak sensitivity to the armature length L and the strong 1" dependence implies that the arc height is more important than the arc length in raising the temperature. Local pinching of the arc and concentration of the current can then give local melting. In the examples of the previous paragraph, the copper will reach its melting point at 0.63 m away from a 6 mm wide armature if the current is increased by 5%. In the case of the 5 cm long armature, a 40% change in current will be required. The thin armature results can be used to model lateral pinching of the arc into a sheet of current. In reality current filaments form spots, not sheets of current. However, many filaments moving together, as are sometimes observed, might be approximated by a sheet. We calculate the temperature rise in the case of the thin armature with constant current and velocity. Remarkably simple analytic results will be obtained. We substitute the asymptotic form of the current in Eq. (15) into Eq. (28) and obtain d m . zyxw We have changed variable of integration where T ' = in Eq. (27) from t' to (' = z - x o ( t ' ) , with z o ( t ) = v t , and we have let I' = I:, where the subscript s indicates that this is the current per height through a sheet. The upper limit of zero comes from the fact that there is very little current ahead 386 zyxwvutsrqponmlkj zyxwvutsrqp zyxwvutsr zyxwvutsrqp zyxwvutsrqp zyxwvutsrqpo zyxwvutsr of the armature, so a point begins to receive heat essentially only after the armature passes it, for t’ > t,, or ( < 0. For the domain where the current has been passing for a time, that is, for < 0 and IrCo(l >> 1, with y fixed and small, the integral can be approximated still further. As (’ varies over its domain, the quantity 6’ T’ stays nearly equal to zero, and all the exponentials are then equal t o one. Deviation from this approximation occurs only in the neighborhood of 6’= 0. The integrand can now be approximated by replacing the exponential with a step function. The resulting integral is then + This is easily evaluated to be From Eq. (30) we see that the isotherm with temperature T, is the straight line with equation If/ = i1e x p [--Tm]. *PC POC2 For copper, the coefficient in this expression is about =Pc = 1.1 x 10-l’ “CmZ/AmpZ. The remarkably simple expression in equation (31) has several interesting consequences. The amount of surface melting can be estimated from it if we neglect the latent heat of melting. If the temperature T , is set equal to the melting point of the rail, we can find the depth to which a given current sheet will melt the surface. For copper, and a current sheet I: = 300,000 Amps/cm of rail height, the slope, ly/zI, of the melting curve is found t o be 3.0 x lo-‘. The mass of material melted will then be presented only numerical results for the temperature at the surface. Conclusion The calculations above have been done for an idealized geometry where there are no variations in the z direction. These simplifications were introduced in order to be able to treat the time dependent problem analytically. The formulas obtained, however, should be of value in understanding the performance of railguns. We believe that despite the simplifications, reasonable estimates of rail currents and temperatures can be made using our methods. We emphasize one of the most interesting results of this paper. This is the possibility of local current density reversals when d l ’ l d t becomes negative. This is simply an inductance effect. The current reverses direction to try to prevent a decrease in the value of B,(z,y,t) in the rail. The same reversal can occur in parts of the armature in a real case. The reversal in the armature does not occur in our model because we assume a known current density distribution in the armature and rigorously compute the current in the rails. In the real problem the current distribution in the armature is not specified; only the total current is. Then a decrease in the total current would decrease the magnetic field in the interior of the armature. The current density in the rear of the armature could easily reverse then, just as it did near the rail edge in Fig. 2, in order to try to keep up the value of the magnetic field in the interior of the armature. The portion of the armature where the current is reversed could then be subject to a magnetic force directed toward the breech. This would have a powerfully disruptive effect in the case of a plasma armature. References J. F. Kerrisk, “Electrical and Thermal Modeling of Railguns”, IEEE Trans. Mag. MAG-20,399 (1984). J. F. Kerrisk, “Current Diffusion in Rail-Gun Conductors”, Los Alamos National Laboratory Report LA-9401MS (June 1982). zyxwvutsrqpo zyxwvutsr [z ] m = -pR 4 2hexp p 0 y T m where p is the density, R is the length of the rail, and h is the rail height. The exponential dependence of the melted mass, m, on the rail parameters should be noted. Taking I: = 300,000 Amps/cm, p = 8.9 gm/cms, R = 400 cm, h = 1 cm, and T, = 1,083 “C, we obtain m = 5.2 grams for copper rails. The large amount of melting is due, of course, to the large value used for the current sheet. We remark that the solution of (30) is singular at the rail surface, y = 0. This singularity is due to the neglect of heat conduction, and to the infinite current density that occurs in the case that the armature has zero thickness. As has been noted by Barber”, this sort of singularity occurs when heat diffusion is neglected in the problem of the sudden turn on of the magnetic field at the plane boundary surface of a semiinfinite space. The singularity does not appear in the case of an armature of nonzero length L,where the current density at the surface is given in (28). Due to the motion of the armature the results are complicated, and in the case of finite L we have R. A. Marshall, “Current Flow Patterns in Railgun Rails”, IEEE Trans. Mag. MAG-20, 243 (1984). G. C. Long, “Railgun Current Density Distributions”, IEEE Trans. Mag. MAG-22,1597 (1986). P.A. Drake, and C. E. Rathmann, “Two-Dimensional Current Diffusion in an EML Rail with Constant Properties”, IEEE Trans. Mag. MAG-22, 1448 (1986). F. J. Young, and W. F. Hughes, “Rail and Armature Current Distributions in Electromagnetic Launchers”, IEEE Trans. Mag. MAG-18, 33 (1982). H. Knoepfel, “Pulsed High Magnetic Fields”, American Elsevier, New York, 1970. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products, Academic Press, New York, 1980, equation 3.954.2. ibid. equation 8.451.6. zyxwvutsr J. P. Barber, “The Acceleration of Macroparticles and a Hypervelocity Electromagnetic Accelerator”, Ph. D. Dissertation, The Australian National University, 1972.