Linear Mass Accelerators

NEW: Study of armature eddy currents and field concentrating shields. (20060704)

Material available for download



Last modified on


Title: “A Brief Theoretical Study of Reluctance-based Mass Accelerators and Actuators”

Mathematical description of the electromechanical system dynamics. Based on a Lagrangian formulation.

Document has been reworked to include saturation nonlinearity and a continuum-based solution method like the finite-element method. It's a bit mathematical, but do not hesitate to read and comment on the conclusions offered after every section.



Title: “Numerical Modeling of a Single Stage 'Sucking Coil' Mass Accelerator.”

Document that describes the numerical solution of the dynamical equations for the electromechanical system described in Part I of the theoretical paper. A hard-copy listing of the numerical code is also included.


IEEE Trans. Mag. paper

Paper published in the IEEE Transactions on Magnetics. Describes the basic model.

The full reference is:

A simple unified physical model for a reluctance accelerator
Slade, G.W.; Magnetics, IEEE Transactions on Volume 41, Issue 11, Nov. 2005 Page(s):4270 - 4276



Title: “Notes on FEA Modeling of Coilgun Mass Accelerator

This draft document presents the theory behind modeling an axisymmetric coilgun using the geometrically versatile finite element method.

(I have restored the original version, because a disk crash and a work overload has prevented me from completing the rewrite. Keep an eye out for the new version, which will include notes on using the new versatile finite-element modeler: BigShot1.0)



Title: “Notes on the effects of metallic coil sheaths and armature losses in reluctance coilguns”

This is a draft document that describes a simple study of the effects of field concentrating shields and armature eddy currents.


Another IEEE Trans. Mag paper

This paper treats the finite element solution of the reluctance coilgun where coil and material losses are present. Here I explain how I extend the use of the magnetisation model within a finite-element framework. Good stuff (I hope!)

The full reference is:

Notes on a Fast FEM Solver for a Reluctance Mass Accelerator

Slade, G. W.; Magnetics, IEEE Transactions on Volume 42, Issue 9, Sept. 2006, Page(s):2184 - 2192.


Motion.c and associated files (TAR)

Source code, Unix Tarball


Motion.c and associated files (ZIP)

Source code, Zipfile


Motion_ver2.c and associated files (TAR)

Source code, Unix tarball. Version 2. Contains simulations for various driver situations, e.g., SCR, dump-and-quench and halfbridge drivers. See the README file here for more information.


Motion_ver2.c and associated files (ZIP)

Source code, Zipfile. Version 2


Motion_ver3 and associated files (TAR)

Source code, Unix tarball. Version 3. Contains simulation code for a saturable armature. SCR, halfbridge and dump-and-quench driver topologies are included. See the README for more details.


Motion_ver3 and associated files (ZIP)

Zipfile of Version 3 code.


Updates to page


Date of modification

Simulation of non-saturable core


Release of first numerical code (nonsaturable core, text-based command-line code.


Proposed experimental coil


Reworked theoretical description to include core saturation. Proposed driving circuit for initial experiments.


Fixed formulation for Part II in theoretical article. Also, constructed first ferrite sheathed launch coil.


Scrapped...yes...scrapped Part II entirely and rewrote from first principles. Now seems to make sense. Also, constructed and tested coil driving circuit based on IGBT.


Some pictures of the first generation single-stage launcher. Driving circuit is included. Also, have a look at a simplified semi-quantitative discussion of energy-transfer efficiency in the mass accelerator.


Have a look at a brief HOWTO on constructing the accelerator coils and the quenching resistor.


Some pictures and experimental results for the two-stage accelerator are presented here.


A collection of useful circuits can be found here.


New FEM based model developed. More to come..


Why in the world did I do this?

This started as a bit of musing on a rainy weekend and has resulted in a brief theoretical exploration of the effect presented in this PDF file.

These devices make excellent physics demonstrators which illustrate the conversion of electrical energy to mechanical energy (and vice-versa). Some visionary scientists (and non-scientists) propose that similar devices could be used to launch material into orbit from the surface of planets or moons where there is no atmosphere. (Note that the maximum kinetic energy of the armature in reluctance based accelerators will be limited by saturation effects. No escape velocity-capable devices are discussed here!) More down to earth uses include all sorts of actuators in automobiles, manufacturing machines and household appliances.

One of the problems in designing devices of this type is their low efficiency in converting electric to mechanical energy. I hypothesize that this is a four-fold problem based on 1.) eddy-current losses in the armature, 2.) retarding forces caused by currents on the armature, 3.) resistive losses in the electric system and 4.) poor timing of current switching which requires large dissipation in the switching elements.

In order to improve efficiency, we need to understand the physics of the electromechanical system. This theoretical study is a start. The next step will be to assemble a numerical simulation of these equations and optimise the system parameters to enhance efficiency. (Note that the model here does not yet include armature currents. This is reasonable if the armature is composed of laminated or non-conductive powdered iron. A further development will include currents in the armature, since these currents are likely to be a major degrader of efficiency.)


26 Oct. 2004... I have managed to get a numerical model of accelerator to work. I am in the process of documenting and verifying it. At the moment, I have verified that total system energy is conserved (to almost within machine precision) during a complete simulation. This is a very good sign that all is well with the simulation. Furthermore, the exchange of energy between the electric and mechanical system is clearly visible. (See the graph of system variables at the bottom of the page.) The simulation shown in the graph used the following physical parameters:



Coil length

10 cm

Armature Length

14 cm

Effective gap length

0.2 cm

Capacitor value


Armature diameter


Number of turns in coil

1000 t

Coil resistance


Armature mass

9 g

Capacitor initial voltage

55 V

Coil initial current

0.0 A

Initial armature velocity

0.0 m/s

Initial armature position

0.0 m

Computational time step


Tolerance for Newton method


Briefly introducing the simulation code, it uses a Crank-Nicolson time-stepping scheme (accurate to second order) with a Newton-Raphson iteration to solve the set of non-linear algebraic equations that arise from the time-discretisation of a transformed set of four first-order differential equations. Convergence of Newton's method is usually fast (2-4 steps) and up to 105 time steps have been used.

It should also be mentioned that I have corrected a few small mistakes in the theory document (the PDF at the beginning of this page). I discovered the errors when I was constructing the code....

The normalised force on the projectile with constant current in the coil. Projectile length is 1.5, coil length is 1.0.

Plot of system variables for coilgun described in preceding text. Core is entering coil from time = 0ms to approximately 10ms. The “force-free” period is from 10ms to 11.75ms. The core is exiting the coil from 11.75ms to nearly 16ms. After 16ms, the simulation terminates because all the interesting system behaviour has finished.


28 Oct. 2004 -Wooweee! the documentation on the mass accelerator program is ready. There is also a program listing at the end. For convenience, the source code is also available for download, so you can experiment with it. It is available as a ZIP file and a TAR file. Unpack the source code and compile it with “gcc -O3 Motion.c Coil.c gauss.c -o Motion -lm” (without the quotes, of course...and assuming you have the Gnu C compiler on you machine). The executable is “Motion”. Run the code and enter your system parameters. The output appears on the standard output and can be piped into a postprocessor or redirected to a file for further processing or plotting (all this assumes a Unix-like system, It should work under Windoze too, if you have a C compiler.) This output is arranged in columns in the following order

Time (sec) Volts(V) Current(A) Armature velocity(m/s) Armature Position(m) and Energy(J)

Take care when entering the data at the input stage. All units are in MKS (meter-kilogram-second) form.


15 Nov. 2004 - Two new things:

  1. I am reworking the theory and numerical model to include the effects of armature saturation. The working assumptions include: using a piecewise linear B-H curve, only armature core suffers saturation and, in the linear inductance region, the permeabilities are still infinite (good approximation since high permeabilities are assumed). When saturation (Bsat) is achieved, the armature is assumed to possess a constant magnetic polarisation density M. It is not possible to use the simple inductance model here.

  2. To assist in verifying the models, an experimental launcher will be constructed. The magnetic materials used in the shield are Amidon FT-87A-W and FT-140A-W ferrite toroids. The proposed armature will use the Amidon R33-050-300 ferrite rod. A (nearly) scale drawing is shown below. More details as construction and testing progress!

  3. Note on ferrites in general: The saturation flux density is around 4500 gauss (0.45 T). A back-of-the-envelope calculation based on the dimensions of the toroids indicates that the flux density within the sheath material will be about 25% of the core flux (because the shield has four times the cross-sectional area of the core). If the flux density is 0.45 Tesla in the shield, there will be about 1.8 tesla in the core - a large field, indeed. A rough calculation yields a current of about 300A for this flux density (assuming 180 turns in the coil).


23 Nov. 2004 - Reworked the formulation such that saturation of the armature can be included. See the “Brief Theoretical Study....” document for details. The next stage in the modeling work will be to include the saturation in the numerical model. More on this later.

On the experimental front, I would like to try a type of “dump-and-quench” topology for driving the coil. The overall efficiency may be degraded by the use of the resistive quench circuit. However, the velocity of the exiting armature can be maximised because the quench circuit will (hopefully) reduce the coil currents to negligible values as the armature exits the coil. The figure below gives the general circuit topology.

Note that the impulse delivered to the armature can be controlled by combinations of Vdd and drive pulse width. Quench speed is controlled by the value of R. It is also possible to use a quenching capacitor, but this will cause different dynamics to appear in the circuit. If a capacitor is used, it is possible to recover the energy for use in the next firing of the coil (but adds to circuit complexity).


Dec 15, 2004 - Finally made corrections to the formulation for the saturated core. Initial numerical simulations indicate that system energy is properly conserved. (Also, completed a simple proof of energy conservation....will add it to the theoretical document soon. The numerical solution algorithm which accounts for saturable armatures is almost complete.

On a practical side, I have constructed my first coil based on the design with ferrite toroids. Need to build the driving circuit and modify a microsecond timer to measure armature speed. Pictures, technical details and measurements to come.


Jan. 18, 2005 - Finished formulation and posted updated theoretical document. It seems to be OK, but if you spot any possible errors, let me know! Sorry about the intense mathematical flavour of the paper, but I some of us do this for fun! However, I hope some of the conclusions are useful to those whose interests are a bit less mathematical. It was a sometimes frustrating, but finally fruitful exercise for me. I had a great Christmas holiday pondering these questions.

Also, the first coil driving circuit has been completed and tested. Performance seems to be pretty good, but I need to finish the speed-measuring circuit before I can verify any of my predictions or start optimising things. I also hope to put together a couple of optical triggers: one for firing a second accelerator coil (for a two-stage coilgun) and another to use for triggering a camera for some high-speed photos!!!

Circuits and pix are forthcoming...... I promise!


Jan. 24, 2005 - First circuits and pix are here!


March 2, 2005 - Finished two-stage launcher. Carried out initial testing. Still need to tidy up circuit diagrams an put together a cohesive document on simulations and experiments. Also want to construct a current measuring device which tracks the time evolution of the current in the coils.


April 13, 2005 - Completed the basic model with saturation of the projectile. Using a simple algebraic B-H relationship, we can study how the force on the projectile changes as the material approaches saturation. Initial computations are encouraging, in that they seem to follow the trends observed experimentally. (I'm still putting together some of the test gear that I need, so a more detailed experimental validation will have to wait a little bit longer.)

In the meantime, I have posted the source code for you to try out. Please let me know if you think you have found any errors. Also, I would be very happy if someone would like to collaborate on developing a nice graphical interface for these codes! Feel free to contact me..


January 4, 2006 - Finally completed the first version of a finite-element based model for the launcher. It includes the following features:

  1. Fully graphical visualisation of fields

  2. Easy, text-based input

  3. Automatic generation of finite element meshes...no muss, no fuss!

  4. SCR, half-bridge and dump-and-quench driver topologies are available.

  5. Reluctance and Thompson (induction) coilgun models can be simulated.

  6. Armature and sheath saturation as well as conduction currents are included in the model.

As with any early release, there are bugs to work out. However, the code seems to give reasonable results that compare well with experimental data as well as the previous models decsribed in this page. A full version of the documentation should be ready within a couple of weeks. I am still undecided about whether to make the software generally available on this site or to send it to interested users only on request.

Stay tuned!