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A Simple Unified Physical Model for a Reluctance Accelerator G. William Slade, Member, IEEE
Abstract— In this work, we develop an illustrative, physically unified model for treating a solenoidal reluctance linear accelerator. A simplified field-based physical model is developed from basic principles and cast in a variational (Lagrangian) form. The equations of motion are presented in a convenient form for computation, from which a fast and compact numerical model is constructed that also accounts for the effects of core saturation as well as the energy exchange between the field and armature motion. The results of this model are then compared with experimental results obtained from a small-scale mass launcher apparatus. Good agreement between experimental and computed behavior has been observed. Index Terms— Coilguns, Electromagnetic launchers, Linear motors, Reluctance machines, Variational methods
I. I NTRODUCTION
E
LECTROMAGNETIC accelerators and linear motors have attracted significant attention in recent years. Induction coilguns, which rely on the interaction between induced eddy currents on a conductive, non-magnetic armature and the stator magnetic field to generate acceleration, have been proposed for military and space purposes [1]-[5]. Here, however, we focus on the the reluctance mass accelerator because appreciable acceleration of a ferromagnetic slug requires lower currents than in the induction machine, thereby simplifying switching circuitry in the experimental apparatus. Moreover, this device finds use in a range of actuators and electromagnetic tools (e.g. nail-guns). Bresie and Andrews [6] indicate that the relative efficiency of reluctance launchers is good in comparison with induction launchers. Many references make use of the finite element method to compute magnetic fields within a solenoidal coil, into which a ferromagnetic slug is inserted. Forces can be readily computed from the field solution (e.g. by integrating the Maxwell stress tensor), but it is generally difficult to treat the full problem dynamically, because this requires treatment of a moving boundary (although Chang, et al. describe a method for doing this using a hybrid FE-BE method to model an induction launcher [5] - see [7] for another method of treating moving boundaries.) Here, we wish to directly model the exchange of energy to and from the armature during different stages of acceleration. For this reason, we have developed a dynamic model using a unified variational (Lagrangian) approach, which makes use of the familiar energy relationships and a simplified field model, avoiding the immediate need for the finite-element Manuscript received June 10, 2005. Author can be reached at Via per Schievenin 21, Quero (BL), I-32030, Italy; Tel: +39 0439 788 219 or +32 (0)2 537 4217; email:
[email protected]
method. (Note that this is possible by virtue of the “longcoil” geometry that we treat. Launch coils whose length is shorter than or of the same order of the coil diameter will require more rigorous treatment of the magnetic field, e.g. by finite element or boundary element method.) By avoiding the need for finite element analysis, solution time for generating meaningful solutions is reduced. Like Tomczuk and Sobol [8], we choose to ignore eddy currents by modeling and experimenting with armatures where eddy currents are suppressed (powdered ferrite and steelwire/epoxy). Also, instead of explicitly treating reluctance (as in [6]), we use a simple magnetisation model for simulating the effects of saturation. As will be shown, this model makes apparent the clear dependence of armature force on the saturation magnetisation in situations of strong saturation. We believe that this simulation reasonably illustrates the basic physics of the reluctance accelerator, given the chosen set of assumptions. The results illustrate clearly the exchange of energy to and from the armature. For this reason, the full circuit dynamics are observed, particularly at high armature velocities. This provides a realistic picture of the electrical stresses that coil switching devices are subjected to as well as a clear understanding of the mechanical behavior. The theoretical generality of the approach also allows the straightforward inclusion of rigorous, time-dependent field models, particularly those based on Lagrangian methods (e.g. finite element and boundary element methods). II. T HE COIL GEOMETRY It is convenient to start off with a study of a single launcher stage, which is represented schematically in Figure 1. Given this geometry, we can enumerate some of the simplifying assumptions (and their consequences). Here, 1) assume that the longitudinal (z-directed) B field is dominant (radial field is ignored) and constant (with respect to position in the core, not in time), 2) flux density is directly related to magnetic field intensity (and armature magnetisation ) in the armature via an algebraic saturation relationship, 3) the armature material is isotropic, 4) hysteresis is ignored (losses will be underestimated as a result, small error in force/circuit dynamics), 5) coil sheath is assumed unsaturated and of infinite permeability (saturation in sheath is ignored, Amp´er`e’s law integral path non-zero only in core region and in air gaps), 6) armature is longer than the coil stage, 7) circumferential air gap at coil stage ends is modeled represents using an effective gap length , where
2
Moveable core entering coil l z Coil sheath l−z
Coil Windings
2a
III. M ATHEMATICAL FORMALITIES The Lagrangian of an electromechanical system, in a general sense, takes the form
z.
C Capacitor that supplies firing charge Assume a small circumferential air−gap of effective length β here (and on the other side too) Moveable core fills coil l
!
is the mechanical kinetic energy, is the elecwhere " tromagnetic ”kinetic” energy and and are the electric and magnetic potential energies, respectively. The magnetic potential energy consists of the sum of the contributions by the armature-filled and air-filled regions of the& core. If the '% armature, of mass # , is moving with velocity $ , then the mechanical kinetic energy is simply
)( * #
' % +.,
(2)
Electromagnetic “kinetic” energy (which is in effect an interaction energy between the coil current and the magnetic field) is 0/
z
2a
(3)
132547678967: ' #65
O
i J
G A + K
M %
n
%
A + -
@
' J
P
M
i
n i n n
(
i
%
% $
*
G 9
A
K + >
AO +
%
;
O
9
'
i
A; K +
n i n n
(
i
(26) (27)
O
(28)
P9
(29)
$
(30)
These equations are written in a form that highlights the component physical phenomena for clarity of exposition. Note that it is possible to eliminate the first equation by substituting (26) where current appears in the other equations. However, 9 no harm is done leaving the algebraic equation in place in the numerical method. When the firing switch opens, the coil current is “quenched” by the the coil series resistance and an external resistor. Expressions (27) and (29) become %
%
(24)
In this case, the force is negative (slows the armature down). Energy will be transferred from the armature back into the magnetic field, possibly causing the coil current to increase for a short time.
n
O #
(23)
(25)
i J
4
(21) (22)
@
'%
O
M %
K
i
In this case, there is no force on the armature; it experiences no acceleration while in this stage. However, the circuit continues to exhibit non-linear dynamics. When the armature exits the coil, one gets K
In order to solve the equations of motion when the transistor switch is closed, the expressions in (18) - (25) are recast in terms of capacitor voltage , coil current , armature velocity ; $ , flux density and armature position ' . 9 The charge variable M N P is readily transformed by recognizing that . It is ; of differentialstraightforward to arrive at the following set algebraic equations for the case (where the armature is entering the coil):
4
;
4
A O
+
4
K >
(31) 9
(32)
is the resistance of the external “quenching” respectively. resistor. There are a number of ways to treat this problem numerically. Classical Crank Nicolson time stepping is simple to implement and has good conservation properties for energy
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V. A N E XAMPLE C ALCULATION The capacitor in Fig. 4 is assumed to be charged to 100V and the armature starts from rest. When the transistor turns on, the system exhibits the resistor-capacitor-inductor-withmovable-core dynamics described by our system Lagrangian. When the transistor switch opens, the effect of the capacitor is removed and we are left with a resistor-inductor-movingarmature system where the current strongly decays. Table I shows the conditions for the simulation
Ferrite armature Mild steel Perfect non-saturable
25
Armature speed (m/s)
and generalized momentum. It is also less likely than explicit methods to suffer instabilities as a result of changing “stiffness” characteristics of the governing equations as the solution evolves. Also, since the system (26) - (30) is nonlinear, the Newton-Raphson method is used to compute the values of the state variables at each time-step. Since the saturation characteristic is given by a closed-form expression, all derivatives needed by the Newton-Raphson method can be analytically described (this allows a fast computation). See [11] for details of the method.
20
15
10
5
0 0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 Time (sec)
Fig. 5. Results for the velocity for three types of armature during the firing pulse. The effect of armature material saturation is to reduce exit velocity.
250
Ferrite armature Mild steel Perfect non-saturable
TABLE I PARAMETERS USED IN
INITIAL ILLUSTRATIVE SIMULATION .
Value 100 0 0 0 30000 0.4 1.0 136 10.0 30.0 35.0 25.0 4.0 2500
Units volts (initial value) amperes (initial value) velocity (initial value) position (initial value) F ohms ohms turns mm mm mm g mm dimensionless
The saturation flux density was varied from 0.45T for a ferrite armature to 2.0T for a mild steel armature. A perfectly non-saturable armature was simulated to provide a benchmark for the coil dynamics. The velocity of the armature is plotted in Fig. 5. The pulse duration was adjusted for maximum exit velocity for each armature example. Notice that the steel and ideal armatures exhibit a mild deceleration at the end of the firing cycle as a result of residual coil current during the armature exit from the coil. The flattening of the velocity curve at around 3.5msec occurs because the armature is slightly longer than the coil. No net force is exerted on the armature when it occupies the entire core region. The evolution of the current is given in Fig. 6. In the initial stages, the current rise behavior is like that of the air-core inductor, since the armature has not moved very far into the core yet. The current then begins to be limited by circuit series resistance effects after a millisecond. It is at this point that the curves begin to separate. This is because the motion of the armature begins to extract energy from the
Coil current (A)
200
Variable
150
100
50
0 0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 Time (sec)
Fig. 6. Results for the coil current for three types of armature during the firing pulse.
magnetic field at a significant rate, giving rise to an apparent “motional resistance” which manifests itself as a “back-emf” % (which is dependent, in part, on ' ). The easily saturated ferrite armature is clearly less efficient (seen by the gentle slope in the current) at drawing energy from the magnetic field, whereas the steel and ideal non-saturable armature generate significant motional resistance (which causes a steeper roll-off in the current during the time that the transistor switch is on, particularly when the steel armature approaches its maximum velocity. When the transistor switch turns off at 5.5msec for the ferrite armature and around 3.5msec for the steel and ideal armatures (to achieve maximum exit velocity, i.e. when the armature “nose” begins to exit the coil, the transistor is switched off) the coil current experiences an initial steep decay as the field energy is dissipated in the quenching resistor and circuit losses. The decay profile for the ferrite armature is monotonic. However, the ideal and steel armatures cause a
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slight rise in the current when they begin to exit from the coil. If there is still current flowing in the coil when the armature exits, the armature will experience a retarding force, which returns energy to the magnetic field (and hence, the coil current) from the armature kinetic energy. This is also observed experimentally (as will be seen). Furthermore, when all resistances vanish, energy is conserved in the simulation. This is an important verification of physical validity of the numerical model. EXPERIMENTAL CASE STUDY
A small coil launcher was constructed (seen in Fig. 7) for the purpose of verifying the numerical model. Using the system (schematically depicted in Fig. 4), measurements of coil current and exit velocity were carried out. Initial capacitor voltage (before firing) is 104 V. Two armatures, one consisting Z of a bundle of seven 4.5mm diameter mild steel rods ( 2.2T) fixed togetherZusing epoxy as well as another of Amidon ). The length ofC - * mix 33 ferrite ( , n C the projectile was 34mm, the diameterC 12.7mm and the mass was 25g. A series of firing experiments was carried out. The average exit velocity (measured with an optically triggered pulse timer) for the mild steel armature was 18.9 m/sec. Measured electrical-to-mechanical conversion efficiency was approximately 9%. The conversion efficiency is found using the ratio of armature exit kinetic energy to the potential energy drawn out of the capacitors during firing, i.e.
#B$
+ X TmV X
P G ;
[ X+
Z + XUV J ;
TABLE II PARAMETERS USED IN SIMULATION FOR EXPERIMENTAL COMPARISON .
Value 104 0 0 0 40000 0.38 0.7 136 8.0 30.0 35.0 25.0 2.0 2500
250
Simulated coil current Measured coil current
200
150
100
50
(33)
The simulation (whose parameters are given in Table II) predicts an exit velocity of approximately 21.5m/sec. The ferrite exit velocity tended to lie around 8.4 m/s while the simulation predicts an exit velocity of 9.5 m/s. Conversion efficiency is very low in this case: less than 2%.
Variable
Fig. 7. Illustration of single stage launching coil used in experiments. Note the high-permeability powdered iron toroids that make up the sheath enclosing the coil windings.
Coil current (A)
VI. A N
Units volts (initial value) amperes (initial value) velocity (initial value) position (initial value) F ohms ohms turns mm mm mm g mm dimensionless
The simulation can be expected to overestimate the exit velocity somewhat, given that frictional forces, sheath saturation effects (if there are any) and eddy-current losses are ignored. Moreover, it is interesting that such a simple model
0 0
0.001
0.002
0.003 Time (sec)
0.004
0.005
0.006
Fig. 8. Comparison of a simulation with measured values of coil current during firing sequence.
gives reasonable velocities over such a wide range of armature magnetic properties. At this point, it can be said that the keys to improving efficiency seem to include: using armature material with a high saturation field, keeping circuit and winding losses to a minimum and allowing the coil current to die out sufficiently before a gap opens up between the trailing end of the armature and the coil inlet opening. It is interesting to note that the armature initial permeability does not have a great effect on the force exerted (which is seen from the expression for W J N n ( ( ( ). when n ( , n G C In order to further verify that the correct physics are being represented by the model, the current was experimentally tracked over a firing cycle. Fig. 8 shows a plot of simulated current evolution with measured current values represented by points. The measured and simulated current profiles correspond well up to the midpoint of the switch on-time, exhibiting the expected circuit resistance current limiting. After this, there is some divergence which could be attributed to eddy currents, coil end-effects or more complex armature saturation than
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cannot be represented with the simplified field model used here. Overall, the measured current profile follows the trends illustrated by the simulation. It is encouraging that at the end of the firing cycle, when the transistor has switched off, a slight rise in the current is observed (the “hump” between 4 and 4.5 msec). The armature loses a small amount of energy to the magnetic field as a result of the expected retarding force when the armature exits the coil in the presence of residual coil current. Current was also measured for the firing of the ferrite armature. No current “hump” was observed after switch turnoff and the overall current profile was little changed from the air-cored inductance behavior. In fact, it was not possible to generate significant repeatable difference data between the “dryfire” situation and the ferrite firing situation. This is consistent with the simulation results for an easily saturated armature, as seen in Fig. 9, where the ferrite armature firing
250
Dryfire Ferrite armature
Coil current (A)
200
150
100
50
cycle. On the other hand, the steel armature (with its high saturation field) allows velocities where energy transfer from the coil field to mechanical motion is readily enhanced. Based on this observation, it can be seen that a suitable choice of armature material is a crucial starting point for the optimisation of the reluctance accelerator. Furthermore,if the armature enters the coil at the start of the firing cycle with a significant velocity, energy transfer efficiency can be improved further by exploiting the increased “motional resistance” effects as well as reduced coil on-times (with the corresponding reduced energy loss in coil circuit resistance). This indicates a route for optimising the efficiency of multistage launchers. The results obtained in this investigation indicate that the major physical phenomena seem to be reasonably well represented. Indeed, the choice of focusing on the longitudinal magnetic flux density in this particular accelerator geometry seems to produce good simulation results. However, we have not treated saturation effects sheath nor eddy currents in the sheath and armature. We have also ignored hysteresis, relying on the assumption that the energy in any remanent magnetisation is small in comparison to the reversible energy changes in the coil magnetic materials as a result of coil current. These give rise to additional loss mechanisms and perturbations to the system dynamics. The sheath can always be sized such that its saturation is not a significant phenomenon in the overall system. Additionally, eddy currents are minimized by using powdered iron, ferrite, laminates or wire/epoxy core materials. Topics of importance that still need further study include end air-gaps, initial armature position and velocity as well as a set of comprehensive rules for scaling of the armature dimensions and mass with respect to coil dimensions and current.
0 0
0.001
0.002
0.003
0.004 0.005 Time (sec)
0.006
0.007
0.008
0.009
Fig. 9. Coil current during a dryfire (empty coil) and a ferrite armature firing. The difference between the two is very small. This is indicative of weak transfer of energy to the armature.
current time dependence almost coincides with the empty coil current profile. The effects of a small motional resistance can be seen during firing on-time as well as a slight core energy “discharge” effect after the driving transistor switches off (just before 6ms). VII. C ONCLUSION Through the use of a simple mathematical model, it is possible to gain considerable insight into the behavior of the reluctance accelerator. For example, in order to obtain efficient energy transfer, use of an armature with a high saturation magnetisation is needed, whereas the initial permeability is not so critical (as long as it is of order 500-1000). It is interesting to see that the efficiency of energy transfer to and from the armature depends not only on its saturation magnetisation but also on the armature velocity. This indicates that the ferrite armature in this study, because of its weak saturation field, cannot achieve the high speeds which allow the armature to extract energy at a high rate during the later stages of the firing
R EFERENCES [1] B. Marder, “A coilgun design primer,” IEEE Trans. Magn., vol. 29, no. 1, pp. 701-705. Jan. 1993. [2] R. Kaye, “Operational requirements and issues for coilgun electromagnetic launchers,” IEEE Trans. Magn., vol. 41, no. 1, pp. 194-199, Jan. 2005. [3] G. Hainsworth and D. Rodger, “Finite element modelling of flux concentrators for coilguns,” IEEE Trans. Magn., vol. 33, no. 1, Jan. 1997. [4] S. W. Kim, H. K. Jung and S. Y. Hahn, “An optimal design of capacitor driven coilgun,” IEEE Trans Magn., vol. 30, no. 2, pp. 207-211, Mar. 1994. [5] J. H. Chang, E. B. Becker and M. D. Driga, “Coaxial electromagnetic launcher calculations using FE-BE method and hybrid potentials,” IEEE Trans. Magn., vol. 29, no. 1, pp. 655-660, Jan. 1993. [6] D. A. Bresie and J. A. Andrews, “Design of a reluctance accelerator,” IEEE Trans. Magn., vol. 27, no. 1, pp.. 623-627, Jan. 1991. [7] H. C. Lai, D. Rodger and P. J. Leonard, “Coupling meshes in 3D problems involving movements,” IEEE Trans. Magn., vol. 28, no. 2, pp. 1732-1734, Mar. 1992. [8] B. Tomczuk and M. Sobol, “Field analysis of the magnetic systems for tubular linear reluctance motors,” IEEE Trans. Magn., vol. 41, no. 4, pp. 1300-1305, Apr. 2005. [9] S. Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in Communication Electronics, J. Wiley & Sons, 1984. [10] C. Lanczos, The Variational Principles of Mechanics, Dover, 1970. [11] W. H. Press, S. A. Teukolsky, W. H. Vettering and B. P. Flannery, Numerical Recipes in C++, Cambridge University Press, 2002.
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G. William Slade (M’87) earned his Bachelor’s and Master’s Degree in electrical engineering from the University of Maryland, College Park in 1985 and 1990, respectively. He obtained his PhD in electrical engineering from Purdue University, West Lafayette, IN in 1993. In all cases, the author’s studies focused on electrophysics and numerical methods. He was a Visiting Lecturer in electromagnetism and RF/microwave electronics at the Royal Melbourne Institute of Technology, Melbourne, Australia from 1993 to 1994. From 1994 to 2000, he went on to work at the Commonwealth Scientific and Industrial Research Organisation as a Research Scientist under a CSR/Windsor Fellowship where he studied the modeling of heat and mass transfer in high temperature convective and electromagnetic drying of wood. In 2000, he joined the Australian Telecommunications Cooperative Research Center 3G project at Victoria University and in 2001 began work again at the Royal Melbourne Institute of Technology as a Research Associate on analog systems in 3G communications and (later) as a Senior Lecturer in electromagnetics, antennas and high-frequency electronics. Since 2003, he works independently on a range of electronic and information technology projects. Research interests include electromagnetic theory, variational and energy methods in physics and engineering, theoretical mechanics, advanced numerical methods in fluids and electromagnetism, RF and microwave electronics and pulsed power. Dr. Slade is also a member of the Society for Industrial and Applied Mathematics (SIAM). Moreover, in 2001-2002, he served as Chapter Chair of the Victorian Section of the IEEE Microwave Theory and Techniques/Antennas and Propagation Society.
