220 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 39, NO. 1, JANUARY 2011 Capacitor-Driven Coil-Gun Scaling Relationships Zhang Yadong, Wang Ying, and Ruan Jiangjun Abstract—Scaling method is an applied scientific method em- ployed in experimental analysis. Also, in the analysis of an elec- tromagnetic-launch system, the scaling method can be an aid in the conversion of test results from models to original. This paper focuses on the capacitor-driven coil-gun scaling relationships be- tween the original and the model. The current-filament method is used to derive the scaling relationships of a capacitor-driven coil gun. If the input scale factors are chosen properly, output scale factors can be calculated by the scaling-relationship equations. Two single-stage coil guns were constructed and tested to verify the scaling method. The calculated scale factors agreed quite well with the test data. Further, three-stage coil guns were simulated based on Ansoft to verify the scaling method of the field results and changes in velocity with respect to the discharge position. It is shown that the scaling method is feasible. These laws allow, with proper application, the construction of experimental models in which phenomena similar to those occurring in the original are reproduced. The result can then be recalculated, using the scale factor for the physical quantity, into the original configuration. A detailed derivation and validation will be presented in this paper. Index Terms—Coil gun, current-filament model, model, scaling. I. INTRODUCTION COAXIAL capacitor-driven coil gun consists of a barrel A formed by an array of stationary coils, which create a magnetic field for propelling an armature. The coils of the barrel are fed in sequence by a set of capacitor-driven circuits. Com- pared with a rail gun, a coil gun has virtues of high efficiency, no Fig. 1. Filament model of the capacitor-driven coil gun. sliding electrical contact, and suitability for launching massive objects [1]–[4]. stage coil guns were constructed and tested to verify the scaling The scaling method has been widely used in electromagnetic- method. Further, finite-element simulations based on Ansoft launch technology [5]–[7]. Due to resource constraints, con- were constructed to verify the scaling relationships between the ducting experiments in models makes economical sense. two models, including circuit and field results. Model-system test results and the experiences gained during the fabrication and testing can be extremely valuable. However, II. DERIVATION OF THE COIL-GUN how to extrapolate the result from model tests to original SCALING RELATIONSHIPS constructions is more essential. As a bridge, the scaling method In a coil gun, forces are characterized by the current and can set up the relationships between the model and the original. the variations in the mutual inductance between the armature Once scaling relationships are established, the performance and coils. The mutual inductance could be determined using of the original construction can be reproduced by experiment the current-filament method described in [8]. It assumes an on the scaled model. The result can be recalculated, using axisymmetric accelerator geometry. The set of equations can be the scale factor for the physical quantity, into the original derived by considering the induction acceleration of a circular configuration. This paper, based on the current-filament method filament, driven in a repulsion mode by another concentric and some electromagnetic-field equations, deduced the scaling circular filament subjected to an arbitrarily varying voltage. relationships of a capacitor-driven coil gun. Then, two single- An induction coil gun based on the current-filament method is modeled by the equivalent circuit shown in Fig. 1. Manuscript received January 10, 2010; revised May 9, 2010; accepted Employing Kirchhoff law, the network equations of the ith June 1, 2010. Date of publication July 15, 2010; date of current version stator coil is expressed as January 7, 2011. The authors are with the School of Electrical Engineering, Wuhan University, dI i−1 dI m d Wuhan 430072, China. R I + L ci + M ck + (M I )=U Color versions of one or more of the figures in this paper are available online ci ci ci dt ccki dt dt caij aj i at http://ieeexplore.ieee.org. k=1 j=1 Digital Object Identifier 10.1109/TPS.2010.2052266 (1) 0093-3813/$26.00 © 2010 IEEE YADONG et al.: CAPACITOR-DRIVEN COIL-GUN SCALING RELATIONSHIPS 221 where J current density, t t time, 1 σ electrical conductivity, Ui = Ui0 − Ici dt, t ≥ ti. (2) Ci μ permeability. t i As a prerequisite, the model must be subjected to physical Employing Kirchhoff law, the network equations of the jth processes similar to those existing in the original. The equation filament is expressed as systems for the original and the model are the same and differ only in the indexes of the physical quantities. We denote as dI i d R I + L aj + (M I ) () and (a) the coordinates associated with the original and aj aj aj dt dt ackj ck K k=1 model, respectively. Scale factor used to describe the scaling m () = K () B = K B dI relationship is defined as , i.e., B , etc. All + M ak =0. aajk dt (3) homologous values of the original and the model must, for each k=1,k=j quantity, maintain a constant proportionality to each other. With this notation, the equations for the model can be written as The basic coil-gun equations are expressed as n m i−1 dM KI dIck acij K K R I +K ck M +··· F = IciIaj = ma (4) Rci Ici ci ci Mccki ccki dz Kt dt i=1 j=1 k=1 m K dI KI d v = adt (5) K Ici L ci +K aj (M I )=K U Lci K ci dt Mcaij K dt caij aj Ui i t t j=1 l = adtdt. (6) (1a) t K K 1 The resistance equation is expressed as Ici t KU Ui =KU Ui0 − Ici dt (2a) i i0 K C Ci i R = ρl0/s. (7) ti K i d The self-inductance and mutual inductance are K K R I +K Ick (M I )+··· Raj Iaj aj aj Mackj ackj ck Kt dt k=1 L ∝ f(μ, x),M∝ f (μ, x). (8) m KI dI K dI K aj L aj +K Iak M ak =0 Laj aj Maajk aajk The efficiency of a coil gun is calculated as Kt dt Kt dt k=1,k=j 1 m v2 m v2 η = 2 0 = 0 . (3a) n n (9) 1 2 2 n m CiU CiU K K K 2 0 0 Ici Iaj Macij dMacij i=1 i=1 K F = I I =K K ma F K ci aj dl m a l i=1 j=1 The electromagnetic-field equations for the system can be written as (4a) B Kvv =KaKt adt (5a) ∇× = J (10) μ J ∂B 2 ∇× = − Kll =KaKt adtdt (6a) σ ∂t (11) ∇·B =0 K K ρl (12) ρ l0 0 KRR= (7a) Ks s where K L∝KμKxf(μ, x),KM M ∝KμKxf (μ, x) Ui voltage applied to the ith coil; L (8a) 2 2 Rci resistance of the ith coil; KmKv mv Kηη = n (9a) Raj resistance of the jth armature; K K2 C U 2 Ci U0 i=1 i 0 Lci self-inductance of the ith coil; KB B Laj self-inductance of the jth armature; ∇× =KJ J (10a) KxKμ μ Mccki mutual inductance between the ith and kth coil; K J K ∂B Maajk mutual inductance between the jth and kth armature; J ∇× =− B (11a) Mcaij mutual inductance between the ith coil and jth KxKσ σ Kt ∂t filament; KB I i ∇B =0. (12a) ci current in the th coil; Kx Ick current in the kth coil; Iaj current in the jth filament; Physical similarity entails the condition that the equations Iak current in the kth filament; for the original and the model differ only by a constant. This B magnetic flux density, means that the power products of the scale factors must equal 222 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 39, NO. 1, JANUARY 2011 Fig. 2. Original model and scale model of the experiment. each summand in (1a)–(12a). For example, the conditions on the scale factors of (1a) can be expressed with the following: Fig. 3. Geometry parameters of the coil-gun experiment. (a) Geometry of the original. (b) Geometry of the model. K K KI K K =K Ick =K Ici =K aj =K . Rci Ici Mccki Lci Mcaij Ui Kt Kt Kt (13) TABLE I SCALING RESULTS OF THE EXPERIMENTS With this notation, one could obtain the condition equations for the scale factors from (1a)–(12a). Because a coil-gun system is often constructed by linear materials, the permeability and electrical conductivity of the model and the original are the same. We could set Kρ = Kσ = Kμ =1as the preconditions. Finally, the scaling relationships can be transformed and sum- marized as (14). The solutions of these equations give the scale laws which govern the relationships of the scale factors with each other for the physical quantities. If a coil-gun model is constructed to research the original, the scaling relationships of (14) must be followed, where Kt must be equal to the square of the geometric scale factor Kx, and the gun current-peek scale factor KI must be equal to KU0Kx, etc. The original and the model have the same efficiency. The scale factors could be selected flexibly based on different experimental conditions. The results gained from experiments on the model can now be converted into the original ⎧ ⎪ Kη = Kρ = Kμ = Kσ =1 ⎪ K = K = K = K =1/K ⎪ v R Rci Raj x ⎪ K = K = K = K = K = K = K = K ⎪ M L r l1 l2 r0 z x ⎪ 2 ⎨ Kt = Kx K = K3 Ci x (14) ⎪ K = K = K ⎪ B Ui U0 ⎪ K = K = K = K = K K ⎪ Ick Iaj Iak Ici U0 x Fig.