Coilgun Acceleration Model Containing Multiple Interacting Coils

Coilgun Acceleration Model Containing Multiple Interacting Coils Kurt A. Polzin,∗ Amanda B. Cipriano,† and Adam K. Martin‡ NASA-George C. Marshall Space Flight Center, Huntsville, AL 35812 Connie Y. Liu§ Georgia Institute of Technology, Atlanta, GA 30332 A coilgun operates by pulsing current through an axially-arranged series of independently-controlled coils inductively interacting with a small, electrically-conductive, azimuthally-symmetric projectile to accelerate it to high velocities. The electrical circuits are programmed to pulse current through the coils in such a way so as to impart further electromagnetic acceleration in each stage. A method is developed to calculate the mutual inductance between the coils and between each coil and the projectile. These terms are used to write a system of first-order ordinary differential equations governing the projectile velocity and the current flow in each coil. While the inclusion of the electromagnetic interactions between coils significantly complicates the equation set as more coil sets are included in the problem, casting the problem symbolically in mass matrix form permits solution using standard numerical Runge-Kutta techniques. Comparing a projectile with a single-turn to that comprised of nine-turns, the inductance of the former is much smaller, but this leads to a greater induced projectile current. The lower inductance and greater current appear to offset each other with little difference in the acceleration profile for the two cases. For the limited cases studied, coils with a discharge half-cycle equal to the time for a projectile to transit from one coil to the next yield increased efficiency. I. Introduction ULSED electromagnetic accelerators and launchers have been investigated because of their ability to propel pay- Ploads to high velocities, with significant effort having been expended studying a capacitor-driven concept known as a coilgun accelerator.1–4 In a coilgun (see schematic, Fig. 1), energy in a series of capacitors is discharged in sequence through inductive coils, producing a time-varying magnetic field according to Faraday’s law that induces op- posing currents in a conductive projectile and interacts with those induced currents to electromagnetically accelerate the projectile via the Lorentz force. The coilgun differs from the well-known railgun concept in that the acceleration in the former is accomplished entirely through electromagnetic field interactions between two (or more) separate current loops, while the accelerating projectile, or armature, in a railgun is required to complete the single high-current electrical discharge path. In recent years, the coilgun has been investigated as a means of in-space electric propulsion. Systems of this type are of interest because, unlike other electric propulsion systems that must generate plasma, these inductive accelerators employ small cylindrical aluminum projectiles called macrons. The use of an already-conducting body alleviates the energy requirement to heat and ionize the propellant in plasma accelerators. In proof-of-concept testing, macrons massing 2 g have been accelerated in a multistage coilgun to the neighborhood of 400 m/s, and it is estimated that 5, 6 these systems could produce a range of specific impulses (I sp) from 600-1000 s. While electrothermal thrusters can operate in this regime, the frozen-flow losses associated with a thermal acceleration process can lead to lower efficiency relative to electromagnetic acceleration in a coilgun. A coilgun-based propulsion system possesses the traditional advantages of using a pulsed system, such as the ability to operate using variable power levels and throttle the average thrust through adjustment of the pulsing rate. 5 ∗Propulsion Research Scientist; Associate Fellow AIAA. †NASA Pathways Intern ‡Physicist §Graduate Research Assistant 1of11 American Institute of Aeronautics and Astronautics separate coils CL projectile direction of acceleration Figure 1. Schematic image of a multi-stage coilgun (after Ref. [1]). The solid macrons also store as a solid as opposed to a high-pressure gaseous or liquid propellant. There are no physical or electrical connections to the projectile resulting in minimal barrel wear and degradation, with the reduction in frictional heating potentially permitting operation at higher repetition rates relative to railguns. 5, 7 The coils in a coilgun and/or the projectile can be comprised of multiple turns of conductor, yielding a greater mutual inductance between the two conducting loops and permitting greater transfer of magnetic field energy into acceleration of the projectile. In the instance of multiple, independently actuated acceleration coils, projectiles can be accelerated using a prearranged sequence of timed pulses to achieve greater velocities than what can be realized using only a single coil. The timing of this pulsing sequence can be better controlled using solid-state switching components, potentially increasing the power processing and energy recovery efficiencies. 6 There has been significant experimental and modeling research on optimizing coilgun designs, 7–11 but analysis and optimization of the coilgun geometry has proven to be a relatively complex undertaking. This is due to both the highly coupled nature of the equations governing the acceleration process and the electrodynamics of the system, and the time-varying nature of the mutual inductance between the coils and the projectile. Recent work has resulted in the identification of nondimensional scaling parameters, permitting a more systematic exploration of the parameter space of possible operating conditions.7 Owing to analytical difficulties, the modeling efforts to-date have been focused primarily on systems consisting of multiple, independent, non-interacting coils operating upon a projectile. However, the coils in a coilgun are typically close enough that if they are operating simultaneously, they should interact not only with the projectile but also with each other through the magnetic field. This situation of coils interacting with each other through a mutual inductance between each other makes the modeling significantly more complicated. The writing of a coupled set of circuit equations and an equation of motion for projectile acceleration in a multiple-stage coilgun is an involved but still analytically tractable problem. A difficulty arises in quantifying the mutual inductance between pairs of conductive coils and between the coils and the projectile as a function of projectile position. In this paper, we describe a method by which these terms can be determined for a system consisting of any number of coils and then as an illustration, present solutions to the model for a system consisting of five coils simultaneously interacting with one projectile. We solve the governing equations for varying accelerator stage capacitance and energy to demonstrate the effects varying these parameters have on acceleration and efficiency. This is, to our knowledge, the first work where the fully-coupled system of circuit equations was solved in the modeling of a coilgun. In section II we present a finite element magnetic field modeling method for the multiple-stage coilgun configuration. We use this method to calculate the self-inductance and pairwise mutual inductance for each conducting body in the three-stage coilgun configuration. The acceleration model for a projectile interacting with an arbitrary number of independently-operating but electromagnetically-coupled coils is given in section III. Calculated results obtained by using the acceleration model are presented in section IV for two different projectile types, while further discussion of the work and potential future research is given in section V. II. Inductance Modeling The self-inductance of each coil and the mutual inductance between the coils and between each coil and the projectile as a function of projectile position were calculated through finite element magnetic field modeling with a 2-D axisymmetric magnetostatic solver (QuickField, Tera Analysis, Los Angeles, CA, USA). The program formulates the problem as Poisson’s equation for a vector magnetic potential A, where the magnetic field B = ∇×A, with either Dirichlet (used in this work) or Neumann boundary conditions. Poisson’s equation is solved over an unstructured mesh through a successive over-relaxation method. The solution converges when the change in field energy between 2of11 American Institute of Aeronautics and Astronautics all dimensions mm 5-turn 5-turn 5-turn coil #1 coil #2 coil #3 φ 2.06 2.06 projectile 2.0 11.83 10.0 7.53 r CL z z =0 z =9.18 z =18.36 direction of (a) projectile motion 1.0 9-turn projectile r C L z (b) Figure 2. Two-dimensional axisymmetric schematic (not drawn to scale) of the finite element model used to simulate the magnetic field and compute the inductance profile for a system of three coils and one projectile (dimensions modeled from the experiment in Ref. [6]) where the projectile is modeled as (a) a single turn cylinder and (b) a 9-turn solenoid. (The projectile is at z =0when it is symmetric about the the dashed line bisecting coil #1.) iterations at every point in the domain is below a user-defined threshold percentage. If the convergence threshold cannot be met during an iteration, the grid mesh is automatically refined. A portion of the coil-projectile system employed in this study is schematically shown in Fig. 2 and is based on the five-turn coil geometry in Ref. [6]. In the present work, only 3 coils need to be

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