PACS №: 52.90+z
B.M. Novac and I.R. Smith
Department of Electronic and Electrical Engineering
Loughborough University,
Loughborough Leicestershire,
LE 11 3TU, UK
A Zero-Dimensional Computer Code for Helical
Flux-Compression Generators
Contents
1. Introduction
444
2. Basis of the Numerical Code
444
3. Inductance Calculations
445
3.1. Calculation of Li (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
3.2. Calculation of Mij (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
3.3. Calculation of dLg /dt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
4. Resistance Calculations
447
5. Other Losses
448
5.1. Voltage Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
6. Code Calculation Layout
448
7. Typical FCG Results
448
8. 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
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
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
444
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)
dL
dI
+I
+ R(t)I = 0
dt
dt
(1)
where L(t) = Lg (t) + Ll and R(t) = Rg (t) +
Rl (t), with Lg (t) and Rg (t) being the generator
inductance and resistance, and Ll and Rl (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.
3.1.
Inductance Calculations
Calculation of Li (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
"Electromagnetic Phenomena", V.3, №4 (12), 2003
in these paper are given in International System Units,
with µ0 being the magnetic permeability of free space)
Lhelix = µ0 πN 2 r2 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 2dimensional 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 rcoil surrounding a coaxial armature of outer
radius rarm (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
2
2
)
− rarm
Lsection = µ0 πN 2 (rcoil
× F (rcoil − rarm , 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 Lg (t) with Ns sections
determined from
Lg (t) =
Ns
X
i=1
Li (t) +
Ns
X
Mi,j (t)(1 − δi,j )
(4)
i,j=1
where Li (t) is the effective self inductance of section
i, Mij (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
445
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 Kcorr to the section effective
inductance can be based on the simple geometrical
formulae Kcorr = 1 for Fig. 1a, Kcorr = 1 − V1 /V0
for Figs. 1b to 1e and Kcorr = 0 for Fig. 1f, where
V1 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 V0 represents the
volume shown in white (the volume between the coil
and an unexpanded armature).
446
3.2.
Calculation of Mij (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
1
(L12 − L1 − L2 ),
2
1
= (L123 + L2 − L12 − L23 )
2
M1,2 =
(5a)
M1,3
(5b)
where Lij and Lijk are hypothetical sections of length
zij and zijk equal to the sum of the length of sections
i and j or i, j and k, respectively: zij = zi + zj and
zijk = zi + zj + zk .
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]
"Электромагнитные Явления", Т.3, №4 (12), 2003 г.
A Zero-Dimensional Computer Code for Helical Flux-Compression Generators
(a)
(a)
(b)
(b)
Fig. 2. MARK IX characteristics: a) load current (—
) fast code prediction, (•) experimental results [15];
b) dI/dt (—) fast code prediction, (•) experimental
results [15].
Fig. 3. FLEXY characteristics: a) load current (—) fast
code prediction, (•) experimental results [13]; b) dI/dt
(—) fast code prediction, (•) experimental results [13].
3.3.
2
2
µ0 πN1 N2 (rcoil
− rarm
)
2z1 z2
2
×[z12
F (rcoil − rarm , z12 ) − z12 F (rcoil − rarm , z1 )
−z22 F (rcoil − rarm , z2 )], (6a)
2
2
µ0 πN1 N3 (rcoil
− rarm
)
M1,3 =
2z1 z3
2
F (rcoil − rarm , z123 )
×[z123
M1,2 =
2
+z22 F (rcoil − rarm , z2 ) − z12
F (rcoil − rarm , z12 )
2
−z23 F (rcoil − rarm , z23 )], (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.
"Electromagnetic Phenomena", V.3, №4 (12), 2003
Calculation of dLg /dt
This calculation requires the inductance Lg (t) to
be calculated at a large number of time points ti (a
time increment of δt = 10 ns is normally adequate)
between t = 0 and t = tf , where tf is the time
when the armature/coil contact point reaches the
end of the stator coil. A close approximation to
the time variation of Lg is given by dLg (ti )/dt =
(Lg (ti+1 ) − Lg (ti ))/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) = RDC (t)T C(t)SKIN (t)P ROXY (t)
(7)
where Rdc (t) is the dc resistance, T C(t) is the
temperature correction of the conductivity and
447
B.M. Novac and I.R. Smith
(a)
Fig. 4. Ranchito characteristics: load current (—) fast
code prediction, (•) experimental results [16].
SKIN (t) and P ROXY (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.
5.1.
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
448
(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 Rdc (t) files. An existing
MATHCAD integration method is then used to
"Электромагнитные Явления", Т.3, №4 (12), 2003 г.
A Zero-Dimensional Computer Code for Helical Flux-Compression Generators
Table 1.
Final load
Type
1)
Energy
multiplication
Effective
Max/min
explosive
charge2) (kg)
internal
voltage3) (kV)
EBS
(kV/cm)
Name [ref]
Load (nH)
current
(MA)
Large
Mark IX [15]
56.5
23.5
25
45.4
155/40
155
Medium
FLEXY [13]
38
7.8
26
14.2
60/15
22
60/20
92
4)
Medium
Ranchito [16]
335
1.6
60
Medium
HEG-24 [17]
340
5.4
225
13.0
150/115
450
Medium
EF-3 [18]
110
3.0
60
11.0
84/50
55
Medium
HEG-16 [17]
550
1.8
500
5.8
155/120
600
Medium
EF1-b [18]
1100
0.67
70
4.4
135/50
145
Mini
HEG-8 [17]
270
0.62
250
0.74
60/50
800
Mini
5)
55
0.3
2.4
0.128
84/20
520
TTU [19]
13.3
Mini
Lyudaev [20]
20
0.65
6.3
0.070
7/2
83
Micro
SANDIA [21]
100
0.013
4.4
0.010
15/7
500
1) 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 dLg /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/cm3 .
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. 2–5. In all cases for
which the experimental dI/dt signal was available,
"Electromagnetic Phenomena", V.3, №4 (12), 2003
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
449
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 dLg /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 COMAGIII [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.
[1] Altgilbers L. et al., Magnetocumulative
Generators – N.Y.: – Springer-Verlag. – 2000.
[2] Novac B.M. and Smith I.R., Classification of
helical flux-compression generators // Proc. MGVIII, Tallahassee, Florida. – 1998.
[3] Wheeler H.A. Simple inductance formulas for
radio coils // Proc. Inst. Radio Eng. – 1928. –
V. 16. – P. 1398–1400.
[4] Grover F.W. Inductance Calculations – New
York: Dover. – 1973.
[5] Kalantarov P.L. and Teitlin L.A. Inductance
Calculations. – Bucharest: Tehnica. – 1958.
[6] Lundin R. A handbook formula for the
inductance of a single-layer circular coil // Proc.
IEEE. – 1985. – V. 73. - P. 1428–1429.
[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. – 1997. – V. 15. – P. 379–395.
[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): 2dimensional simulation of helical generators //
Proc. Pulsed Power Plasma Science Conference.
Las Vegas, Nevada. – 2001. – P. 110–113.
[10] Knoepfel H. Pulsed High Magnetic Fields –
North-Holland, Amsterdam and London: – 1970.
[11] Butterworth S. Alternating current resistance of
solenoidal coils // Proc. Roy. Soc. – 1925.
– V. 107.
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
450
References
[12] Arnold A.H.M. The resistance of round-wire
single-layer inductance coils // Proc.IEE. – 1951.
– V. 98. – P. 94–100.
[13] Novac B.M. et al., Design, construction and
testing of explosive-driven helical generators //
J.Phys.D:Appl.Phys. – 1995. – V. 28.
– P. 807–823.
[14] Neuber A.A. et al. Thermodynamic state of the
magnetic flux-compression generator volume //
Proc Pulsed Power Plasma Science 2001. Las
Vegas, Nevada, USA. Eds. Reinovsky R. and
Newton M. – P. 98–101.
"Электромагнитные Явления", Т.3, №4 (12), 2003 г.
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. – 1989. – P. 475–478.
[16] Cash M.A. et al. Benchmarking of a flux
compression generator code // IEEE Trans.
Pl.Science.
– 2000. – V. 28. – P. 1434–1439.
[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. – 1980. – P. 641–649.
[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. – 1990. – P. 403–410.
[19] Neuber A. et al. Electrical behaviour of a
simple helical flux compression generator for code
benchmarking // IEEE Trans.Pl.Science. – 2001.
– V. 29. – P. 573–581.
[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. – 1994. – P. 607–618.
[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.
– 1980. – P. 163–180.
[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. – 1980. – P. 205–218.
"Electromagnetic Phenomena", V.3, №4 (12), 2003
451