A Zero-Dimensional Computer Code for Helical Flux-Compression Generators

Transcription

1 PACS : z B.M. Novac and I.R. Smith Department of Electronic and Electrical Engineering Loughborough University, Loughborough Leicestershire, LE 11 3TU, UK Contents A Zero-Dimensional Computer Code for Helical Flux-Compression Generators 1. Introduction Basis of the Numerical Code Inductance Calculations Calculation of L i (t) Calculation of M ij (t) Calculation of dl g /dt Resistance Calculations Other Losses Voltage Breakdown Code Calculation Layout Typical FCG Results Conclusions 450 Abstract Although several complex computer programmes available in the literature provide detailed information on both the overall characteristics and internal magnetic and electric fields of FCGs, they usually involve considerable computing time. However, the present interest in the use of FCGs as the energy source for high power microwave and other equipment brings about the need for a fast code for use in the preliminary stages of a system design. The paper describes a programme that is much faster than the detailed programmes, but nevertheless provides an accurate prediction of the terminal characteristics. A step-by-step procedure is used to calculate the different inductance variations, and both skin and proximity effects are taken into account in determining the resistances. Theoretical predictions are compared with available experimental data for several familiar generators and the results are analysed. The most important parameters for a high-energy gain design as suggested by the fast code are highlighted. 1. Introduction There is presently considerable interest worldwide in the development of compact and single-shot expendable pulsed-power sources for use in microwave generators, aviation technology, outer space and oil drilling. The flux compression generator (FCG) [1] is a natural contender to provide the high power/energy requirement of these sources. It is clear however that for much of the associated development to be successful a thorough understanding of the basic techniques of design, manufacture and testing is required. The most basic need is for a simple, fast and 444

2 A Zero-Dimensional Computer Code for Helical Flux-Compression Generators reliable numerical model for use in designing FCGs with helical, multi-section stator coils. Although the literature contains numerous descriptions of methods for calculating the time variations of both the inductance and resistance of these generators, very few of these (and paradoxically not the simpler ones) present sufficient information to enable a code to be developed. The purpose of this paper is to present in full detail a very simple technique for modelling a FCG, which nevertheless still provides an accuracy of prediction that is sufficiently high for design purposes. The resulting code is very fast, and enables tens or even hundreds of runs to be performed in limited time to investigate fully the parameter space available when producing a new FCG design. Results provided by the code are discussed and conclusions are drawn. 2. Basis of the Numerical Code The numerical modelling described below is based on the voltage-balance equation L(t) di dt + I dl + R(t)I = 0 (1) dt where L(t) = L g (t) + L l and R(t) = R g (t) + R l (t), with L g (t) and R g (t) being the generator inductance and resistance, and L l and R l (t) being the load inductance and resistance. The generator arrangement is considered to be conventional [2], with the diameters of both the coil and the armature remaining constant along the working length and the load current I returning through the armature. The coils of the multi-sectioned stator winding can each have a different (but constant) pitch, and the number of turns in each section can be chosen arbitrarily. As the code is intended primarily for use in applications that require generators ranging in size from micro to medium size (as defined in [2]), any effect of magnetic pressure on the armature dynamics is ignored. For convenience, the code is written in MATHCAD 2001 but it can be easily implemented in any other language such as MATLAB, FORTRAN, TURBO PASCAL or C++, with possibly even higher speeds of calculation. 3. Inductance Calculations 3.1. Calculation of L i (t) An essential prerequisite of any fast and simple generator code is a simple formula for the inductance of a helical constant-pitch coil, having the length z, inner radius r and N turns. Fortunately a very good approximation to this is available in [3] as (all formulae avaible from i.r.smith@lboroac.uk in these paper are given in International System Units, with µ 0 being the magnetic permeability of free space) L helix = µ 0 πn 2 r 2 F(r,z) (2) where F(r,z) = 1/(z + 0.9r) for a long coil (z > r) and F(r,z) = 10/(11z + 8r) for a short coil (z < r). The accuracy of these formulae was confirmed for a number of widely different geometries, by comparison with results provided in handbooks [4,5], other formulae known to be accurate [6] and 2- dimensional filamentary modelling [7]. Although the differences could be up to 4 % in extremely short coils, they were normally less than 1 % for the coil designs likely to be used in FCGs. Based on a suggestion of C.M. Fowler et al [8], equation (2) can also be used to calculate the inductance of a helical coil of radius r coil surrounding a coaxial armature of outer radius r arm (i.e., a section of a multi-section helical generator), if it is assumed that the return current through the armature mirrors that in the coil. The resulting formula is L section = µ 0 πn 2 (r 2 coil r 2 arm) F(r coil r arm,z) (3) were again confirmed by 2-dimensional modelling [7]. In addition, separate calculations were made using handbook data to estimate separately the inductances of a number of helical coils and armatures (for the paths of the mirror currents) and the corresponding mutual inductances. When these results were combined to give the inductance of a coil-armature system (a section) the accuracy of agreement with results provided by equation (3) was about the same as in the previous comparisons with equation (2). It should be noted that although equation (2) is carefully formulated to maintain continuity of the inductance variation while moving between short and long coils, this is obviously not so when equation (3) is used. The discontinuity thereby introduced represents the only drawback of this technique, and it appears as sharp jumps in calculations of the time variations of the effective inductance as the length of a section reduces. Since this can easily be overcome by using a simple mathematical smoothing technique, equation (3) was adopted in the FCG code, with the effective initial self inductance of a generator L g (t) with N s sections determined from N s L g (t) = L i (t) + i=1 N s i,j=1 M i,j (t)(1 δ i,j ) (4) where L i (t) is the effective self inductance of section i, M ij (t) is the mutual inductance between sections i and j and δ(i,j) is the Kronecker delta. As the armature expands, the volume between it and the coil is reduced, and strictly speaking "Electromagnetic Phenomena", V.3, 4 (12),

3 B.M. Novac and I.R. Smith Fig. 1. The six possible positions of an armature inside a helical generator section during the magnetic compression process. Volumes shown in white are filled with high magnetic energy density. The magnetic energy density in the grey and black volumes is neglected in the inductance calculations. The grey volumes are shown only for convenience, in order to explain the inductance calculation technique. equation (3) becomes invalid. The exact calculation of this effect on the section inductance requires numerical techniques that are insufficiently fast for the present application, and an alternative approach based on energy considerations was therefore adopted. Apart from the small region close to the armature/coil contact point, the magnetic energy density is distributed relatively homogeneously inside any section of the generator [9]. Thus, as the armature expands within the section, and the magnetic flux is concentrated towards the load, the effective inductance is assumed to be reduced in proportion to the volume between the coil and the armature that is then occupied by the field. Fig. 1 shows all the six possible geometries that can arise in a section, depending on the expansion angle of the armature and the length of the section. For each possibility of Fig. 1, a correction factor K corr to the section effective inductance can be based on the simple geometrical formulae K corr = 1 for Fig. 1a, K corr = 1 V 1 /V 0 for Figs. 1b to 1e and K corr = 0 for Fig. 1f, where V 1 represents the volume shown in grey (calculated as the difference between the truncated volume of the expanded armature and a cylinder representing the initial volume of the armature) and V 0 represents the volume shown in white (the volume between the coil and an unexpanded armature) Calculation of M ij (t) The technique adopted to calculate the mutual inductance between any two sections of the FCG coil is based on the only two possible cases: when the two sections are either adjacent (e.g. section 1 and section 2) or are separated by one or more other sections (e.g. section 1 and section 3). It is then easy to demonstrate, based on energy conservation, that for the case of sections 1,2 and 3, all having the same pitch M 1,2 = 1 2 (L 12 L 1 L 2 ), M 1,3 = 1 2 (L L 2 L 12 L 23 ) (5a) (5b) where L ij and L ijk are hypothetical sections of length z ij and z ijk equal to the sum of the length of sections i and j or i,j and k, respectively: z ij = z i + z j and z ijk = z i + z j + z k. Equations (3) enable each of the inductances in equation (5) to be calculated, and if the cone is expanding within any of them (see Fig. 1), the results need to be corrected using the technique described above. However, in a multi-section generator, the sections have different pitches, and a very close approximation to the mutual inductance is obtained by use of the formulae [5] 446 "Электромагнитные Явления", Т.3, 4 (12), 2003 г.

4 A Zero-Dimensional Computer Code for Helical Flux-Compression Generators (a) (a) (b) Fig. 2. MARK IX characteristics: a) load current ( ) fast code prediction, ( ) experimental results [15]; b) di/dt ( ) fast code prediction, ( ) experimental results [15]. M 1,2 = µ 0πN 1 N 2 (r 2 coil r2 arm) 2z 1 z 2 [z 2 12F(r coil r arm,z 12 ) z 2 1F(r coil r arm,z 1 ) M 1,3 = µ 0πN 1 N 3 (r 2 coil r2 arm) 2z 1 z 3 [z 2 123F(r coil r arm,z 123 ) z 2 2F(r coil r arm,z 2 )], (6a) +z 2 2F(r coil r arm,z 2 ) z 2 12F(r coil r arm,z 12 ) z 2 23F(r coil r arm,z 23 )],(6b) When some of the sections considered have their volume reduced by the armature cone their corresponding terms in equations (6) have again to be corrected. (b) Fig. 3. FLEXY characteristics: a) load current ( ) fast code prediction, ( ) experimental results [13]; b) di/dt ( ) fast code prediction, ( ) experimental results [13] Calculation of dl g /dt This calculation requires the inductance L g (t) to be calculated at a large number of time points t i (a time increment of δt = 10 ns is normally adequate) between t = 0 and t = t f, where t f is the time when the armature/coil contact point reaches the end of the stator coil. A close approximation to the time variation of L g is given by dl g (t i )/dt = (L g (t i+1 ) L g (t i ))/δt which can subsequently be smoothed numerically. 4. Resistance Calculations The resistances R(t) of the coil and the armature of the FCG are calculated simply from R(t) = R DC (t)tc(t)skin(t)proxy (t) (7) where R dc (t) is the dc resistance, TC(t) is the temperature correction of the conductivity and "Electromagnetic Phenomena", V.3, 4 (12),

5 B.M. Novac and I.R. Smith Fig. 4. Ranchito characteristics: load current ( ) fast code prediction, ( ) experimental results [16]. (a) SKIN(t) and PROXY (t) are corrections for skin and proximity effects. The techniques to calculate the DC resistance were explained in [13]. The formulae for the temperature correction are taken from [10] and require the calculation of the relevant Joule energy deposited per unit mass. Skin depth and proximity effect corrections are introduced by using the results in [11] and [12], arranging these as spreadsheets and using interpolation techniques to determine intermediate data. The frequency at any time needed in obtaining these corrections is approximated by di/dt/i. 5. Other Losses The various losses present in FCGs, in addition to Joule heating, have been explained elsewhere [13]. In the present code many of these are introduced in a straightforward manner in order to minimise the calculation time. The code takes into account nonlinear diffusion at the armature-coil contact area, 2π clocking (when the allowed eccentricity is less then 0.1 mm) and voltage breakdown Voltage Breakdown This is often regarded as the most important source of loss occurring in a FCG, and the studies presented later show that it may even exceed the Joule heating loss. The mechanism of the breakdown has been well described elsewhere [13], and the corresponding model was easily implemented into the present code. It is important to appreciate that the electrical breakdown strength (EBS) of the materials in the contact region is unknown. Recent measurements [14] have suggested that the EBS for gases trapped in the very small volume close to the contact point is close to that at STP. The breakdown voltage however depends not only on the EBS of the (b) Fig. 5. HEG-24 characteristics: a) load current ( ) fast code prediction, ( ) experimental results [17] b) internal voltage ( ) fast code prediction. gas but also that of the coil insulation and the quality of the expanded armature surface. These effects are virtually impossible to quantify, and consequently the EBS constitutes the only adjustable parameter in the present fast code. The examples given below, for very different designs, reveal the values that are necessary to match predictions to experimental results. 6. Code Calculation Layout The full set of input needed for the code is: coil (i.d.) and armature (o.d.) diameters, armature thickness and material (copper or aluminium), expansion angle and detonation velocity, crowbar radial position and the complete data for each coil section i.e., length, number of wires in parallel, cable o.d. and insulation thickness. The initial current is given or can be calculated from the initial energy source parameters. Calculations commence with the creation of L(t), dl(t)/dt and R dc (t) files. An existing MATHCAD integration method is then used to 448 "Электромагнитные Явления", Т.3, 4 (12), 2003 г.

6 A Zero-Dimensional Computer Code for Helical Flux-Compression Generators Table 1. Type 1) Name [ref] Load (nh) Final load Effective Max/min Energy EBS current explosive internal multiplication (MA) charge 2) (kg) voltage 3) (kv/cm) (kv) Large Mark IX [15] / Medium FLEXY [13] /15 22 Medium Ranchito [16] ) 60/20 92 Medium HEG-24 [17] / Medium EF-3 [18] /50 55 Medium HEG-16 [17] / Medium EF1-b [18] / Mini HEG-8 [17] / Mini TTU [19] 5) / Mini Lyudaev [20] /2 83 Micro SANDIA [21] / ) The classification is given in [2]. 2) The effective explosive charge is defined here as the mass of the explosive inside the armature for a length equal to that of the helical coil or, if it is the case, the length of the combined helical/cylindrical coaxial assembly. The Table is organised according to this quantity. 3) Maximum internal voltage corresponds to the highest value of the term dl g/dti. Many designs have the lowest values of this term in the first or in the last section. In such cases the minimum internal voltage shown in the Table corresponds to the maximum value predicted during the compression of that section. 4) The explosive mass had been estimated assuming an initial mass density of 1.8 g/cm 3. 5) The TTU generator is certainly able to provide a much higher energy multiplication but for lower initial/final currents. The example chosen here is because of the only accurate data available. integrate eqn (1) and to determine the current/time relationship. The code not only can provide the complete history of the load current, di/dt and internal voltage, but also the wire and armature temperature in each section and can quatify the importance of various types of losses in a particular design. 7. Typical FCG Results This section analyses, with the aid of the fast code, the output performances of some helical, conventional designed [2] helical generators for which sufficient information was available in the open literature. The final aim of this study was to try to understand what parameters are essential for a successful design. For each generator, an EBS value was found for which the final (maximum) predicted load current matches the experimental value. Typical examples are given in Figs In all cases for which the experimental di/dt signal was available, very close agreement was obtained, showing that the code predictions are credible. Such examples are given in Figs. 2b and 3b. The findings, together with certain basic design parameters for the generators investigated, are given in Table 1. Complete information for all the generator designs can be found in the relevant references. In the case of Mark IX [15], the experiment with the lower final current (23.5 MA) was chosen, since for even higher output currents (30 MA [15]) other phenomena could disturb the compression geometry. For example, at very high current levels the magnetic field pressure can lower the armature expansion angle or even give rise to an exotic electro-mechanical loss generated by the very high torque appearing on the armature and producing a rotation. The simple fast code is not intended to deal with such complicated situations. Not all the generators in Table 1 have been developed to maximise the energy multiplication. In the cases of Mark IX [15] and of FLEXY [13] for example, the very high current required "Electromagnetic Phenomena", V.3, 4 (12),

7 B.M. Novac and I.R. Smith by their specific applications meant a substantial reduction of the internal voltage, thus influencing their energy multiplication figure. This was because of the necessity to spread the current over a higher number of paralleled conductors to maintain a reasonable current density. This however lowered the dl g /dt and correspondingly the voltage and energy gain. In the case of the TTU generator [19], a simple design for basic physics studies, and the Lyudaev generator [20], a first stage of a dynamic transformer system, energy multiplication was certainly not a design issue. However, the internal voltage values and the EBS figures are useful (as the type of cable insulation is given in both cases) and the experimental data allows the predictions to be compared with experimental data for the interesting case of minigenerators. As a general conclusion, it is evident from the fast code results that a design to maximise the energy multiplication has to rely on very high values for EBS and a smooth armature expansion, to minimise possible internal electrical breakdowns [13]. Figures as high as 500 kv/cm, and even higher, were predicted by the fast code for some of the generators. It is interesting to note that in the case of the SANDIA micro-generator [21], this very high figure is in agreement with an EBS of 400 kv/cm concluded in a numerical study some decades ago, using COMAG- III [22], the most sophisticated 2D code available at that time. As indicated in Table 1 and revealed by the fast code calculations, a high EBS value by itself is insufficient for high-energy gain. This requires a careful multi-sectional approach to the coil design, in order to maintain throughout the compression time a high and flat internally generated voltage characteristic. According to Table 1, the best design in this respect appears to be the HEG-24 generator [17], for which the predicted internal voltage is given in Fig. 5b. 8. Conclusions A fast code for helical generators having a conventional design had been developed and used in the study of various generator designs. The importance of the cable insulation properties has been illustrated. As the fast code also reveals, the only way to obtain a very high energy gain (multiplication) is to use a large number of sections in the design of the helical coil, to control and maintain a high value flat internal voltage characteristic during the whole compression time. The fast code is used at Loughborough as an essential tool in the initial stages of the designing process of high performance helical generators. Manuscript received August 1, 2003 References [1] Altgilbers L. et al., Magnetocumulative Generators N.Y.: Springer-Verlag [2] Novac B.M. and Smith I.R., Classification of helical flux-compression generators // Proc. MG- VIII, Tallahassee, Florida [3] Wheeler H.A. Simple inductance formulas for radio coils // Proc. Inst. Radio Eng V. 16. P [4] Grover F.W. Inductance Calculations New York: Dover [5] Kalantarov P.L. and Teitlin L.A. Inductance Calculations. Bucharest: Tehnica [6] Lundin R. A handbook formula for the inductance of a single-layer circular coil // Proc. IEEE V P [7] Novac B.M., Smith I.R., Enache M.C. and Stewardson H.R. Simple 2D model for helical flux-compression generators // Laser and Particle Beams V. 15. P [8] Fowler C.M., Caird R.S. and Garn W.B. An introduction to explosive magnetic flux compression generators, Report LA-5890-MS, 1975 (unpublished) [9] Novac B.M. and Smith I.R. Explosive-driven pulsed-power generation program (MURI): 2- dimensional simulation of helical generators // Proc. Pulsed Power Plasma Science Conference. Las Vegas, Nevada P [10] Knoepfel H. Pulsed High Magnetic Fields North-Holland, Amsterdam and London: [11] Butterworth S. Alternating current resistance of solenoidal coils // Proc. Roy. Soc V [12] Arnold A.H.M. The resistance of round-wire single-layer inductance coils // Proc.IEE V. 98. P [13] Novac B.M. et al., Design, construction and testing of explosive-driven helical generators // J.Phys.D:Appl.Phys V. 28. P [14] Neuber A.A. et al. Thermodynamic state of the magnetic flux-compression generator volume // Proc Pulsed Power Plasma Science Las Vegas, Nevada, USA. Eds. Reinovsky R. and Newton M. P "Электромагнитные Явления", Т.3, 4 (12), 2003 г.

8 A Zero-Dimensional Computer Code for Helical Flux-Compression Generators [15] C.M. Fowler and R.S. Caird, The Mark IX generator // Proc.7th IEEE Pulsed Power Conf., Monterey, Ca., Eds. Bernstein B.H. and Shannon J.P P [16] Cash M.A. et al. Benchmarking of a flux compression generator code // IEEE Trans. Pl.Science V. 28. P [17] Chernyshev V.K. et al. High-inductance explosive magnetic generators with high energy multiplication. Megagauss Physics and Technology Ed.Turchi P.J. N.Y. and London: Plenum Press P [18] Ursu I. et al. Pulsed power from helical generators, Megagauss Fields and Pulsed Power Systems, Eds. Titov V.M. and Shvetsov G.A. N.Y.: Nova Science Publ P [19] Neuber A. et al. Electrical behaviour of a simple helical flux compression generator for code benchmarking // IEEE Trans.Pl.Science V. 29. P [20] Lyudaev R.Z. et al. MC-generators with magnetic flux trappers (dynamic transformers). Megagauss Magnetic Field Generation and Pulsed Power Applications, Eds. Cowan M. and Spielman R.B. N.Y.: Nova Science Publ P [21] Gover J.E. et al. Small helical flux compression amplifiers. Megagauss Physics and Technology, Ed.P.J.Turchi. N.Y. and London: Plenum Press P [22] Freeman J.R. et al. Numerical studies of helical CMF generators. Megagauss Physics and Technology, Ed.P.J.Turchi, N.Y. and London. Plenum Press P "Electromagnetic Phenomena", V.3, 4 (12),