William H. Arnold, Stuart Bowen, Kevin Fine, David Kaplan, Margaret Kolm, Henry Kolm, Jonathan Newman, Gerard K., O'Neill and William R. Snow
A mass driver is an electrical device used to accelerate payloads of any material to a high velocity. Small vehicles (called "buckets") containing superconducting coils carry the payloads. These buckets are accelerated by pulsed magnetic fields, timed by information on their position, and are guided by induced magnetic fields set up in a surrounding guideway. Upon reaching the correct velocity, the buckets release their payloads, then they are slowed for recirculation to be reused. A rationale is presented that leads to a relatively simple and near-optimum design, as well as basic programs for calculating acceleration. The transverse oscillation frequencies are found to be invariant to guideway transverse dimensions. A detailed optimization of system mass and a discussion of all structural questions are presented in other papers of this summer study.
INTRODUCTION
The theoretical development of mass-driver designs has intensified since this concept was first published by O'Neill (refs. 1,2). Much recent work is contained in unpublished notes. The three papers on mass drivers in this Special Publication are therefore written to summarize the present body of knowledge; this introduction contains references to the main sources of relevant information presently known.
After the first publication on mass drivers, the next substantial design effort occurred during the 1976 NASA-Ames Study on Space Manufacturing. In that study, there was active participation by professionals with experience in linear synchronous motors and magnetic flight.
Two mass-driver configurations were considered at the 1976 study: planar and axial. The planar system, a flattened, double-sized version of the M.I.T. magneplane (ref.3), had served as the basis for the first mass-driver reference design, partly because calculations could be checked against existing computer programs. The axial configuration (fig.1) was eventually recognized as superior, particularly for reaction engine applications involving maximum exhaust velocity and minimum total system mass. Axial geometry permits tighter coils and therefore higher acceleration for the same total system mass. Basic formulas for mass optimization, general enough to apply to a wide variety of mass drivers (both planar and axial are given in references 4 and 5.
The 1976 study served to verify and extend the basic mass-driver concept, to reveal the advantages of axial geometry, to produce the mass-optimization formulas, and to develop a method for payload guidance after release. It yielded the important result that accelerations of the order of 104 m/sec2 are almost certainly realizable in mass drivers of moderate transverse dimensions.
Building on this foundation, substantial advances were made during the 1976-77 academic year. In a series of four seminars,1 the basic design for a mass driver of high performance (6,000 to 10,000 m/sec) and low system mass (100-200 tons) was developed for application to reaction engines in space.
1 O'Neill, G.K.:Space Flight via Maxwell's Equations, Special Seminar Series, Department of Aeronautics and Astronautics, M.I.T., 1977 (unpublished work).
In the seminar series, the transverse oscillation scaling laws were derived, a simple program for calculating acceleration was written, and improvements were made in obtaining maximum drive for minimum power loss in the drive windings. These results were summarized concisely, without derivations, at the 1977 Princeton/AIAA Conference on Space Manufacturing (ref. 6). In that paper, the concept was applied to an engine used to move large amounts of equipment from low Earth orbit (LEO) to lunar orbit.
During the same period, the first mass-driver model was designed and built at M.I.T.: construction was directed by Henry Kolm and a number of the authors of this paper took part in the assembly and testing. The model, 2 in long with simplified circuits, demonstrated acceleration only; no attempt was made to combine a test of magnetic flight. It operated successfully in the .acceleration range of tens of gravities and was widely demonstrated. Its design is covered in reference 7 and its construction, in reference reference 8. The state of the art of the theoretical design of mass drivers as of September 1977 was summarized at the International Astronautical Federation Conference of that year (ref.9).
EARLIER WORK ON AXIAL SUPERCONDUCTING ACCELERATORS
It has come to our attention that, in addition to the extensive bibliography fisted in reference.5, a "traveling wave accelerator" for superconducting solenoids was proposed and studied earlier by F. Winterberg and co-workers at the Desert Research Institute, University of Nevada, Reno (refs. 10, 11).
In discussing the differences between the mass-driver concept and the Winterberg traveling-wave accelerator, we should note the several key elements of the massdriver system: a linear synchronous motor for acceleration, guidance by magnetic flight, and recirculation of the accelerated vehicle, with only the payload leaving the system. Winterberg's system differed from the mass driver in that he made no provision to maintain synchronization by silicon-controlled rectifier (SCR) switching on the basis of position sensing. Instead, the coils and capacitors were selected to form a lumped parameter transmission line having a local phase velocity that provides uniform acceleration of the traveling wave. This was accomplished by making 
(where L and C are inductance and capacitance per unit length) equal to the correct local velocity. Winterberg showed by approximate analysis that phase stability should be achieved. He also found that resonance (in phase) should be confined to a very small region at the beginning of the track, and that angular instability should be controllable by the relative position of the center of thrust and the center of mass of the accelerated solenoid.
He concluded, however, that there will be inherently unstable transverse translational motion because, in his system, the vehicle always rode the front crest of the traveling step function. In the axial mass driver, the bucket is first attracted and then repelled by each drive coil. Attraction has a centering effect, while repulsion has a destabilizing effect. The net stability is therefore neutral under completely symmetric timing conditions. If, however, the wave form is m, de asymmetric so that the attractive pulse is of larger amplitude than the repulsive pulse, then the net effect will be stabilization of lateral motion. In the latest mass-driver circuit design, it is easy to achieve stabilizing asymmetry by making the dc feeder voltages (capacitor charge) unequal to the extent required.
DESIGN RATIONALE
There are many possible choices in the design of mass drivers, and optimizations will change both as the result of further refinements in design and of improvements in materials (especially superconductors) and components (capacitors and SCRs in particular). We do not wish to halt that process of continual refinement, but we feel that the present state of knowledge is sufficient to justify setting out a "nominal" family of designs, for which the potential user can calculate performance, system mass, and sensitivities to changes in materials, components, and the main variables of payload size, repetition frequency, maximum velocity, and acceleration. Here we consider the electrical design of a particular family of mass drivers (axial geometry, with a specified value for the ratio of bucket to drive coil radii) in the approximation that accelerations are calculated for currents taken to be thin filaments. In reference 12, we consider large-scale structural questions; in reference 13, we present optimization formulas and programs.
Table 1 presents the nomenclature used in all three papers.
| Symbol | Design choice |
|---|---|
| Bucket iB current in each coil, A | 2.5X106D2 |
| mB mass of empty bucket, kg | 296D3 =2.263m1 |
| mBL mass of loaded bucket, kg | 3.23m1 |
| mBC mass of each coil, kg | 74D3 |
| m1 * mass of expelled segments, kg | m1 |
| nC number of bucket coils, m | 2 |
| r radius of bucket coil from axis, m | 0.26D |
| VBC volume of each coil, m3 | 1.634X10-2D3 |
| WB width of square bucket coil, m | 0.1D |
| Drive system ADO cross sectional area of drive winding, m2 | To be optimized |
| Afo cross sectional area of feeder lines, m2 | To be optimized |
| a acceleration delivered to mBL,m/s2 | a |
D mean diameter of drive coil  caliber, m | 0.197m |
| fm microduty cycle, s-1 | 1 |
| fR* repetition (launch) rate, sec-1 | fR |
| i feeder current, A | iD/n2 |
| iD ampere turns peak, each drive winding, ampere-turns | 0.435m1a/D2 |
| Lw single-turn inductance of drive winding including mutual, henries | 2.004X10-6D |
 m inductance length= drive coil spacing, m | 0.185D |
| P phase length or wavelength 4m,m | 0.74D |
| s sector length, m | to be optimized |
| w length of drive winding (circumference), m |  D |
| Nw total number of drive windings per phase | 1.351V2max /aD |
| nw number of simultaneously excited drive windings per sector capacitor with bridging | 2 |
| n2 number of turns each drive winding | - |
| R mean radius of drive coil from axis, m | D/2 |
| Sa acceleration length, m | V2max /2a |
| Sd deceleration length,m | mB/mBLSa=0.694Sa |
| Stot total length of driver, m | 1.694Sa |
| ta acceleration time, sec | Vmax /a |
| V local velocity of bucket, m/sec | V |
| Vmax* launch or exhaust velocity, m/sec | Vmax |
 ratio of bucket and drive coils | r/R=0.52 |
 number of phases | 2 |
 LC ringing frequency (=2 Vp),sec-1 |  |
| max maximum ringing frequency, sec-1 | 8.489Vmax /D |
* Denotes fundamental mass-driver mission variable
Where the notation had already been chosen in earlier publications, we adopted the same notation. For easy reference, table 1 also contains the specific design choices corresponding to our family of designs (discussed below). With our design choices, all main dimensions of the bucket and all transverse dimensions of the drive system are fixed when the drive coil diameter D is chosen (where D is referred to as the "caliber," but it is not equal to the clear diameter through which the bucket passes; that clear diameter is smaller than D because of the finite cross section of the drive coil, necessary insulators or supports, and finite thickness of the guideway). In table 1, several approximations are implicit. The minimum velocity is approximately zero in calculating ta, Sa, Sd, and Stot. Drift space and bucket deflection space (peel-off length) required to separate the bucket from the payload after release are neglected in calculating Stot. As defined, Stot is the correct parameter to use in mass-optimization formulas, but it is smaller than the physical length of the complete mass-driver system. The term Lw is based on a simple calculation of selfinductance and mutual inductance, and does not take full account of the detailed changes in currents in the drive coils adjacent to the coil whose inductance is calculated.
TRANSVERSE SCALING
For our family of designs, we have chosen to leave 
= r/R fixed at 0.52. This choice is arbitrary; a reasoned choice would require knowledge we do not yet have, as we now show.
The magnetic-flight guidance forces are restoring for the two orthogonal position oscillations in the transverse direction and the two orthogonal angular oscillations. To first order, longitudinal motion depends only on the drive currents, and there is no restoring force for the roll mode. A practical mass driver might be wound with bucket and drive coils slightly noncircular to provide a restoring torque for roll.
We chose, in our family of designs, to scale all linear dimensions of the bucket with caliber D, and now calculate the approximate transverse oscillation frequency for D = 5.0 cm. We consider the guideway to be composed of eight strips of equal width and approximate the bucket coil as octagonal (see fig. 2). Assume that a particular current element of the bucket of length y is influenced only by the parallel guideway strip nearest to it. Thus, for each of the eight bucket coil sides, only a single image current is needed to describe the force F felt by that side. When the bucket is centered,
where y = 2D(tan 45)o/2 and 
O
is the permeability constant. To sum the forces on all sides of the bucket when the bucket is displaced a distance x from the centerline position, the restoring force that acts to return the bucket to the centered is
Here half of the contribution to the restoring force comes from sides 1 and 5, while half comes from sides 2, 4, 6, and 8. The corresponding spring constant is 2000 N/m. For s = 0. 3 D/2, leaving a distance 0. 18 D/2 for the finite half-thickness of the drive coils, for insulation, and for the finite thickness of the guide strips, the restoring force for larger displacements is given in figure 3. For bucket mass given by our nominal design, the corresponding oscillation frequency is 37.3 Hz. The notation in this derivation, as in table 1, is that of reference 4.
Consider next the effect on bucket transverse motion caused by scaling mass-driver linear dimensions proportional to D. The dimensions R, r, s, and y all scale linearly with D. The cross-sectional area of the bucket coil is D2 . If it is assumed that the current density in the superconducting bucket coil remains constant, the current flow iB, will also scale as D2 By substituting these scaling factors, the restoring force on the bucket scales as D4.
The spring constant K will vary as D3 since it is the restoring force F divided by the linearly scaling distance s. The bucket mass will also increase as D3. Consequently, the frequency of bucket oscillation, equal to 
will be variant to the caliber D
and so the maximum tolerable angle of bucket motion relative to the axis will also scale as D.
In the absence of active feedback from steering coils placed at intervals along the guideway, the bucket oscillation amplitude would slowly build up as a result of misalignments along the accelerator. There is an extensive literature treating such oscillations, as a result of the 30 years of previous development of particle accelerator theory and practice. At a more sophisticated level of design knowledge on mass drivers, it will be necessary to investigate how the misalignments that drive oscillation buildup scale with D. At our present level, we must close the argument at this point with the comments that we lack sufficient knowledge to proceed further, and that active feedback should be much easier in our case than for particle accelerators: we are dealing with a macroscopic bucket, of fixed shape, whose position can be measured to very high precision by optical scanning. The bucket moves very slowly compared to the velocity of light, so there is ample time to measure, calculate, and feed back correction currents to steering coils located at intervals along the guideway. In reference 13, we show that the mass of a total optimized accelerator changes only slowly as a function of D so long as the mass flow dm1 /dt and vmax are held constant. Therefore, the choice of D does not appear to be critical and can be deferred until a more thorough understanding of misalignments is reached.
Radius Ratio
Tighter coupling would be obtained with a higher value of the radius ratio r/R. However, this would reduce the clearance gap between bucket and guideway. For = 0.52 from ref. 4) and for bucket and drive coil widths of 0.1D, the total allowance for insulation, guide strips, clearance, and thermal insulation surrounding the superconducting coil is 0.14D, or 0.7 cm for D = 5 cm.
It does not seem realistic to reduce clearance by using a much higher value of ; the dependence of the peak gradient of mutual inductance on r/R is shown in figure 4, along with the peak mutual inductance with respect to r/R , which also does not justify a tighter coupling.
Bucket Coil Cross Section
We chose square coils of width 0.1 D. If the coils were much thicker, the drive current pulse could not be timed to optimize the drive simultaneously on all parts of the drive coil, and therefore the efficiency would be lost. Our choice could probably be altered by ±25 percent.
Superconducting Cable Density and Current
We chose 25,000 A/cm2 at an average density of 4.53X 103 kg/m3 , the same values discussed in reference 5. The choice rests on practical operating experience with the M.I.T. magneplane model.Drive Frequency, Coil Spacing, and Energy Transfer
The instantaneous force on one bucket coil due to one drive coil is iB(dM/dx)iD(t), where iD(t) is the value of the ampere-turns in the drive coil at time t (ref. 4). In reference 6, it was found convenient to integrate the drive force over the x region of drive coil excitation, to obtain the total energy change imparted to each bucket coil in traversing each drive coil. During the same interval, the energy loss in the drive coil is proportional to
iD2(t)dt. We therefore obtain maximum energy transfer to the bucket, at minimum power loss in the drive coil, by maximizing [iB(dM/dx)iB(t)dx]/iD 2(t)dt. The problem is slightly complicated by the fact that the induced EMF caused by the moving bucket coil in the drive coil affects diD/dt; the current cycle in the drive winding is therefore not exactly sinusoidal when a charged capacitor is connected to the winding.Although our actual basic drive circuit consists of two drive coils, separated by the spacing of the bucket coils and operated in series from a capacitor, that circuit is equivalent to the simpler one shown in figure 5; that is, a single drive coil with its self-inductance, mutual inductance, and internal resistance, across a capacitor, with an additional generator corresponding to the induced EMF due to bucket motion. The differential equations that describe the current variation in that circuit are
where t is time, dM/dt is the rate of change of the mutual inductance between the total current of one bucket coil and one turn of a drive winding having n2 turns. By the chain rule,
For numerical computation, we put these formulas into difference-equation form with Vc and i as functions of x rather than t and solve for 
i
and c:
for stepwise integration with a step interval x. The velocity v is considered to be constant over the short interval in x over which one drive coil acts-an excellent approximation for all but the first few drive coils of the accelerator. Cast in this form dM/dx is an explicit function of x, independent of n2 but dependent on the caliber D. For purposes of calculation the four constants appearing in the difference equations are called successively a1, a0, a3 and a4.
The calculation can be carried out with high accuracy by use of a program for dM/dx (see appendix A), combined with a difference-equation routine. Alternatively, a good approximation is obtained much more compactly by a simpler program (see footnote 1appendix B). This program carries out the stepwise integration of the difference equations for Vc and i, and yields the total energy transfer to the bucket coil; the program occupies less than 50 program steps and so can be run on even a small programmable calculator.
By use of this simple program, it was found in the paper in (footnote I) and noted in reference 6 that a very good, although not necessarily an ideally optimized, solution is obtained by matching the peak in the i(x) curve to the peak in the curve of dM/dx (fig. 6). If that is done, the sine wave of the drive current is turned on when the bucket coil is a distance 0.37D before the drive coil. We call the distance 0.185D, the "inductance length," m.
We obtain a particularly simple geometry and drive circuit by choosing a spacing between drive coils equal to the inductance length. In that case, successive coils carry currents 90o apart in phase. Only two independent circuits are needed to feed all coils in a given region of the accelerator, one circuit for even-numbered coils, another circuit to feed odd-numbered coils.
Number of Bucket Coils
With progressive refinement of our understanding, we have adopted a drive system in which each coil is separately excited (ref. 5) through its own switches (SCR's). For historical reasons, the number of bucket coils previously chosen was four, in the case of axial symmetry. However, with each drive coil separately excited, we consider anew the question of the optimum number of bucket coils.
For stability against pitch and yaw, the minimum number of bucket coils is two. A possibility (not considered further here) is that one of these two coils might carry most of the total current, the other having low mass and current and intended only to provide stability. Consider two alternative bucket designs, of the same total superconductor mass, divided in one case into four coils and, in the other, into only two coils. To provide equal total energy transfer to the moving bucket, the drive currents required are the same in both cases because that energy transfer is proportional to the product of peak drive current, current per bucket coil, and number of bucket coils. However, for two coils, each drive winding need only be excited for two complete sine waves, while for four coils, four cycles of excitation are needed. Therefore, the power loss is reduced by one-half with two coils, for equal drive-winding mass. Our choice is thus nc = 2.
Feeder losses are also reduced in the two-coil case. For full drive in these drive windings, current must precede and follow each bucket coil. In the notation in ref. 5 the number nw of windings simultaneously excited is reduced from six for four coils to four in the case of two coils. In the section on drive circuit, we reduce nw to two. Figure 7 traces the winding currents as the bucket moves.
Bucket Length
The spacing between the two bucket coils could be any multiple of the spacing between drive coils. We adopt a specific value, with the note that changing to a longer bucket to accommodate a low-density payload would have little effect on bucket mass and no effect on accelerator mass. Given sufficient built-in flexibility in the coil triggering system, it may be possible to accommodate different bucket lengths even in a single accelerator.
Limits on Drive-Coil Spacing
We have chosen for simplicity a two-phase drive system, with the spacing between coils equal to m
. All coils are connected in the same sense, and the bipolar feeder always sends current to two adjacent coils of one phase, which are connected by a common neutral (fig. 8). From the viewpoint of placing the drive currents in the best possible geometric location, drive windings should probably be distributed uniformly, in a thin layer forming a continuous solenoidal coil. That alternative will be studied in the future. We consider here the practical limits on coil spacing set by the simplicity of the circuit we have adopted, but allowing for the possibility of a multiphase drive (more than two phases).
In the mass optimization for the entire mass driver, feeder and capacitor mass can be saved by operating with each drive winding subdivided into n2 turns, with n2 as large as possible. For typical mass-driver parameters, the most serious limit on n2 is set by the mutual inductance MD between drive windings. For
ampere-turns iD, the induced voltage from this source is
(n2MD)
LCiD. This effect increases the SCR standoff voltage by a certain fraction, independent of acceleration.
The fractional induced voltage is approximately inversely proportional to spacing. For an actual multiphase system, it is necessary to sum up the effects of all coils appropriately phased, as summarized below for n2 = 1.
| Drive coil spacing, cm | Total induced voltage as fraction of applied voltage |
|---|---|
| 1.0 | 0.28 |
| .6 | .72 |
| .2 | 2.24 |
| .1 | 4.4 |
Evidently, the induced voltage caused by Current changes in adjacent drive coils limits n2 and also limit the minimum spacing between drive coils. Our choice (equivalent to the 1.0-cm entry) allows moderate value for n2 , of the order of 2 or 4. Much higher values of n2 would result in a much increased standoff requirement for the SCR's.
Payload Dimensions
The bucket length B is chosen as six drive coil spacing (6m = B = 1.11 D). The length of the payload is chosen to be 5/6 of this or 0.925D. To provide a reasonable clearance gap between the payload and bucket coils (to accommodate magnetic shielding, insulation, payload holder, etc.), the radius of the cylindrical payload is chosen as rp = 0.3R. Thus the clearance gap between the outer surface of the payload and the bucket coils is 0.12R = 0.06D. The ratio of bucket length/ diameter is thus 1.95 (taking into account the finite thickness of the bucket coils) and the ratio of payload length to diameter is 3.08.
For a payload of density 
p the caliber D is then
WE assume a nominalp=2X103 kg3.
Drive Circuit
There are two bipolar feeder lines (two split phases) plus one neutral line, and the two feeders are in quadrature, that is, their currents differ in phase angle by 90o. Drive coils are all identical and are connected to the neutral fine on one side, their other side being connected to the four feeders of alternating side and alternating phase (fig. 8). The resulting phasor diagrams are also shown in figure 8. Note that there are only four simultaneously excited coils during the passage of one bucket coil, and that there are always two coils connected in series across one bipolar feeder pair, although the coil connections progress in a leapfrog fashion along the feeder pair. Each coil is fired independently by its own bipolar SCR, which conducts during one entire 360o cycle, whose period is matched approximately to the local velocity by the choice of local LC product.
Bridging SCR's connect adjacent feeder pairs so long as a bucket occupies one of them. This arrangement eliminates the need to double the capacitor energy of individual sector capacitors to allow for the fact that two bucket coils will occupy each sector most of the time. Bridging SCR's allow each bucket to derive energy from two adjacent feeder sectors, and the average conduction distance remains unchanged. Feeder power loss is unaffected, and the number of windings simultaneously driven by one sector capacitor nw is reduced to two.
The ringing frequency of the drive circuit (LC in previous terminology) is chosen to make the coil spacing or inductance length equal to one-quarter wavelength:
SUMMARY
In this discussion of mass-driver electrical design, several small improvements have been made relative to our previous studies. The bucket design is simplified by reducing the number of coils to two. That change reduces winding losses by 1/2 and feeder losses to 2/3 the previous value. Feeders are reduced from three phases to two. Bridging SCR's offer savings by a factor of 2 in the total capacitor energy storage required. Our application of these changes to the reoptimization of total system mass is treated in reference 13. At a later time, it would be worthwhile to consider a wider class of designs than those we treat here. Although it seems unlikely that large improvements are yet to be made, it may well be that our design is as much as 20% away from an optimum system mass.
Next.
