Physical Understanding of Electromagnetic Forming and Acceleration


Simulation of the Thomson Ring Experiment
Analytical description of the electrodynamic and mechanical process while inductive acceleration

1. Introduction
2. Setup of the experiment

3. Modelling of the coil
....3.1 Neumann Equation
....3.2 Inductance Matrix
....3.3 Resistance of the coil
4. Selfinductance and resistance of the Thomson ring
....4.1 Selfinductance
....4.2 Resistance

5. Mutual Inductance between Coil and Ring
....5.1 Mutual Inductance via the Mutual Inductance Matrix
....5.2 Fitting with Multi Exponential Functions
6. Caculation of the acceleration force
....6.1 Via the magnetic field
....6.2 Via Gradient of mutual inductance
....6.3 Table of Equations
7. Simulation of realistic experiments

... 7.1 Software
....7.2 KB1-KB2-FL1,2,3 Experimental Setup
8. Summary and Discussion

9. Literature

Thomson Ring Experiment

On this page I present the theory of electromagnetic acceleration of metal rings via flat/ spiral coils powered by capacitor discharges.
Such a experimental setup is called a Induction Gauss or Coil Gun. I don´t like this name because the goal here is not to create weapons. According to William Thomson (1824-1907) the "Thomson ring experiment" is a common part of experimental physics lectures to explain induction and Lorentz-force. In this sense I want to explain how this experiment may be described with Theoretical Electrodynamics and Mechanics. Theoretical concepts as inductance matrices, mutual inductance, Elliptic Integrals were used to get a system of 8 differential equations. Via Numerical solving of this equations the typical graphs for ring position, speed, acceleration, discharge current, voltage, inductance current, heat temperature and a lot moore could discussed on this page...
This project is just part 1 in the series to describe electromagnetic forming process theoretically.

Max Bigelmayr, May 2015, August 2015 (V1)

1. Introduction

Classical electrodynamics is one of the standart lectures in universities when you study physics [1], [2]. Especially in experimental physics the concepts of induction, lenz law, lorentz force and the maxwell equations are explained via several great experiments. One of these experiments is the famous Thomson Ring Experiment (Fig. 1). A coil with several windings creats a pulsed magnetic field by discharging a capacitor through it- or just using the alternating current from the the socket. The time dependent magnetic field induces an eddy current in a metall ring placed above the coil. The interaction of the current in the ring and the magnetic field yields to an acceleration of the ring away from the coil due the lorentz force. There are many concepts how to explain these phenomena. But no one really tells you how to calculate the currents, forces, speeds in this experiment. Physicists learn how to deal with the covariant maxwell equations, relativistic corrections, retarded potentials and fundamental concepts like the lagrangian density. In this theoretical world concepts as inductance are seen as "not to important". In engineering faculties one may hear some lectures about electromagnetic field theory- but here everything is mostly "very static" with no moving parts. With less physical background the maxwell equations are difficult to understand in any detail, so that many professors only present very easy examples to explain the electromagnetic field theory. So, as you see, the Thomson Ring Experiment is somehow between the subjects physics and electrical engineering.
In 2006 I made several studies on metal forming via pulsed magnetic fields [10]. At this time I had enough experience to do practical experiments, built high voltage capacitor banks, coils and measurement equipment. But I really had not enough mathematical and physical background knowledge to calculate a complex system as the Thomson ring experiment, which is very familiar to the electromagnetic forming technology. I remember many hollidays at my grandmas house trying to combine formulas from old austrian scool books (Schreiner) and books for electrical engineers to find a way to calculate the Thomson Ring Experiment. Everytime when I thought I got the solution I realized that I would have to know the result of an equation to use the eqution (At this time I didn´t know how to handle differential equations). This mades me very crazy and I decided to study physics. Several years later after studying physics I am proud to present a very nice concept to describe and simulate the Thomson Ring experiment here.

2. Setup of the Experiment

At this page I want to present the Thomson Ring Experiment using a capacitor as energy source (Fig. 1). After charging up the capacitor C to a certain voltage U0, the capacitor will be discharged through a flat/spiral coil. In my experiments I mostly used a 150uF capacitor bank charged up to several kV`s. While the discharge of the capacitor (here U=3kV) through the a coil with L=11uH the current reaches peak values around 12kA, which may be approximated by the formula
maximal Diascharge Current Capacitor . [1.1]
The magnetic field of the coil reaches total values around 3 Tesla near to the center (Fig. 2a, b). As the periodic time of a CLR-circle with low damping is described by
Periodic Time CLR Cricle, [1.2]
it´s obvious that an ideal acceleration of the Thomson ring is given when the kinetic energy of the ring reaches it´s maximum while the first half wave of the discharge current pulse:
Acceleration Time[1.3]
The maximal kinetic energy of the ring is
Kinetic Energy Thomson Ring[1.4]
with the efficiency factor k: 0<k<1.


Schematic setup of the Thomson ring experiment


Fig. 1: Schematic setup of the Thomson ring experiment: A capacitor with capacitance C and voltage U0 is discharged through a spiral coil with inductance Lcoil. The magnetic field of the coil flushes through the Thomson ring ( Phi12) and induces an eddy current I2, which rotates in the opposite direction than the cause (rotating current I1 in the coil). The Thomson Ring then creates an own magnetic field, which is aligned contrary to the magnetic field of the coil. These two magnetic fields repel each other, so that the Thomson Ring is accelerated upwards. For an equivalent explanation with the magnetic pressure, watch my old page theory of electromagnetic forming (in German). Later I will present different concepts how to caculate the acceleration force with the lorentz force and gradient of mutual inductance (chapter 6).

Under (unrealistic) linear conditions with constant acceleration the acceleration path is:
Accleration Path Thomson Ring[1.5]
For a given acceleration lenght y one may solve equation [1.4] for vmax and put it into y(t) together with the acceleration time tacc [1.3]:
Accleration Lenght Thomson Ring[1.6]
Solving this for L we get an expression for the magnitude of inductance one should use for the coil:
Approximation Inductance Thomson Ring Experiment[1.7]


  A Thomson ring with mass m=16g is accelerated by discharching a capacitor bank (U=6kV, C=150uF) through a coil. Assuming a acceleration lenght of 4cm the needed inductance of the coil is around
appr. Inductance Thomson Ring Experiment.

The presented assumptions are:
- constant acceleration force
- acceleration time is exactly the first current half wave
- 10% of the electric energy is transferred to kinetic energy
- the acceleration path is known (here 4cm).

For practiacal experiments these assumptions are very difficult and one may have much experience and feeling to take suitable parameter values.

Now let´s discuss the energies in the Thomson ring experiment: A large capacitor/capacitor bank with capacitance C is charged up to a voltage Uo [Fig. 1]. After closing the switch the energy stored in the capacitor will be transformed in the circuit. Energy conservation gives:
Energy Thomson Ring Experiment [1.8]
The total energy in the system is the sum of 7 different energies:
Energy Equation[1.9]
EC(t) is the time dependent stored energy in the capacitor, EL1(t) is the Energy of the magnetic field in the whole circuit 1 , EM(t) is the mutual magnetic energy, EL2(t) is the magnetic energy of the cicuit 2, ER1(t) is the accumulated heat energy of the circuit 1, ER2(t) is the accumulated heat energy of the circuit 2 and Ekin(t) is the kinectic energy of the ring (circuit 2) with mass m. Before doing simulations it´s essential to know all the parameters Lcoil, Lline, Rcoil, Rline, which must be calculated before starting the simulation algorithms. Let´s first start to dicuss the calculation of the coil inductance.

3. Modelling of the coil

3.1 Neumann Equation

The coils used in the Thomson Ring Experiment are commonly in the form of Archimedean spirals. The magnetic field of such coils may be calculated with Biot-Savart law by using suitable parameterisations [12]. Practically it´s not helpful to calculate induction phenomena in metals or metal rings by using magnetic field calculations (Fig. 2b). Of course one could integrate the flux denstiy over the surrounded surface of a certain loop to get the induction current (Fig. 2b). But this requires a lot numerical effort with error propagation due to the many steps one have to follow. Remember the Poisson equation for the vector potential A:
Poisson Equation Vector Potential A [3.1]
This elliptical partial differential equation of 2th order may be solved by using the Green´s function:
Vector Potential [3.2]
Instead of j(r)*d^3r one may write Idr, so that we get for a closed loop:
Vector Potential Ring [3.3]
Surface integration of the magnetic flux density B yields the magnetix flux Phi (unit Wb=T/m^2). Together with the substitution B=rotA we get:
Magnetic Flux throw ring [3.4]
Putting equation [3.3] into equation [3.4] one achieve
Inductance Flux[3.5]
With the definition of L=Phi/I we get the so called Neumann formula (not to be confused with the von Neumann Equation in quantum mechanics!):
Neumann Formula [3.6]

  Spiral Coil Thomson Ring
  Fig. 2a: Current flow throw a spiral coil
Magnetic Field Density Spiral / Flat Coil

Fig. 2b: Magnetic field flux density of a typical spiral coil with 10 windings at a current of 10kA. The magnetic flux components Bx, By and Bz were calculated via numerical integration of Biot-Savart law. The density plot shows the value Bges=sqrt(Bx^2+By^2+Bz^2).


3.2 Inductance Matrix

The Nemann formula may be used to calculated the the self inductance Lii (with i e N) of metall rings as well as the mutual inductance Lij (i=/j) (Fig. 3).
In general one may split up the total inductance Lges of a wire into the inner inductance Lin and outer inductance Lout :

total Inductance[3.7]
Assuming a constant partial inductance we may write
Inner Inductance, [3.8]
and use the arithmetic expression
partial Inner Inductance Wire , [3.9]
which may be deduced by volume integration of the energy density of an infinte long current carrying wire. For a better understanding read the excellent book from E. B. Rosa and F.W. Grover [8], [9]. The outer inductance is the inductive backlash of the conductor itself.
In other words: Imagine a infinitesimal thread instead of a round wire with section A and radius r (Fig. 4). The wire has got the form of a circle with radius R. The thread carries a current Iin. Now imagine a second infinetesimal wire circle with radius R-r in the same plain. The magnetic flux (Phi12=L12*I1) through circle 2 (created by circle 1) correlates directly with the mutual inductane L12, described by the Neumann formula:
Neumann Formula[3.10]
  Mutual inductance between to rings
Fig. 3: Mutual inductance between to rings: The magnetic flux created by the current I1 in ring 1 suffuses ring 2 with the flux Phi12, which is less than the flux Phi1 throw ring 1. Flux. Phi2 may be calculated with equation [3.5]. The mutual inductance L12 between the two rings is defined by the Neuman formula [3.6].

It is obvious, that Lout of a round wire with cross section A, radius r and circle radius R is exact L12 from the imagination with the two infinitesimal threads.
By analytical treatment of the Neumann formula for two rings with circle radius R1, R2 and distance b between the rings one may get the general expression for the mutual inductance L12.
Mutual Inductance between two Rings[3.11]
K(k) is the complete elliptic integral of the first kind, E(k) is the complete elliptic integral of the second kind. The value k is described by the formula:
k Factor Mutual Inductance[3.11b]

For calculating the self inductance of a ring we set R1=R and
R2=R-r according to Fig. 4. Then equation [3.11] becomes:
Outer Inductance Ring[3.12]
k Factor
By splitting up the total inductance we get:
total Inductance Ring
Integration around the circle with equation [3.9] gives 1/4µ0R. With Lout=L12 we get the total inductance of a ring:
total Inductance of a Ring[3.13]
Approximation of this expression gives the expression: (one may find in some engineering books)
Self Inductance of a Circle


Self Inductance of a Ring


Fig. 4: Inductance of a ring: A infinitesimal thread circle (1) with radius R is placed in the middle of the ring section. The thread carries the current Iin, which represents the total current in the ring. Assuming a homogenous current density in the ring, the magnetic field energy may be calculated with equation [3.15]. Then the inner inductance of the ring Lin is described by equation [3.9]. Imagine a second infinitesimal thread circle (2) with the Radius R-r, which is the radius of the inner edge of the ring. The total flux created by (1) may be splitted up into two parts: Part one is the flux through the surface between thread circle 1 and thread circle 2. This flux is the product Lin*Iin. Party two is the flux through circle 2, Phi12, which is nothing other than the outer inductance Lout. The total inductance is described by equation [3.13] or approximated by equation [3.14].

The total energy of a system with several inductances and individual currents may be calculated by integration over the energy density at each point in the space:
total Energy Magnetic Field [3.15]
Another option is to use the summation over all energies and "mutual energies" according to the self inductances and mutual inductances of the cuurent carrying system:
Energy as sum of individual Inductance Circles[3.16]
In a coil the currents are equal in any inductor, so that we get
Currents, [3.17]
ant therefore the total energy is
total energy magnetic Field[3.18]
Comparison of [2.16], [2.17] and [2.18] yields:
Total Inductance Coil [3.19]
So the total inductance of a coil is the summ of all matrix elements of the inductance matrix of the coil. Let´s explain the inductance matrix with an example:
As I figured out in simulations spiral coils with many windings and layers may be approximated well via many rings carrying the same current instead of using the exact form of a Archimedean Spiral.
In Fig. 5 a 3- layer coil with 4 windings in each layer is presented. The section radius of each circle is 1mm. The radius of the smallest rings are 5mm. The distance from one ringe to the next ring is always 10mm. Calculation of all inductances and mutual inductances with equation [3.13], [3.11] gives the inductance matrix for this multi-layer coil (unit is nH in each cell):


Inductance Matrix, Multi Layer Coil


Fig. 5: Model of a multilayer coil with 4 layers: Each layer has got 4 Windings as concentric circles.Via numerical simulation and comparison with real Archimedean spirals one may show that the numeric failure created by this "unreal model" is quite low for "larger winding numbers (N>10)". The radius of the smallest rings are 5mm. The radius of the other sings are 15mm, 25mm, 35mm. The distance between the three layers is 10mm. So the ring to ring distance is 10mm for any neighbor ring. The wire radius of each circle is 1mm. The inductance matrices of the coils L1, L2, L3 are the diagonal elements of the large inductance matrix below. The mual inductances between the layers are L12, L13, L23. Due the inductance matrix is symmetric L12=L21, L13=L31, L23=L32.

Inductance Matrix of a Coil

The summ of all matrix elements (equation [3.19]) is 2695nH. So the total inductance of this multilayer coil is around 2,7µH.
The explained method for inductance calculation may be used for nearly any kind of coils. You may parameterize any wire path and use the Neumann formula [3.10]. When a approximation with circles is possible, just use equation [3.11] together with [3.13] or [3.14]. Then summarize up each element of the matrix with equation [3.19].

3.3 Resistance of the coil - Approximation

Since we use rings in our coil modeling the entire ohmic resistance of the coil is:
Resistance of many Rings [3.20]
The ohmic resistance of one individual ring is roughly:
Ring Resistance [3.21]
with the specific elctrical resistance p, which is 1,7E-2Ohm*mm^2/m for a standart copper wire.

4. Selfinductance and resistance of the Thomson ring
4.1 Selfinductance

The self inductance of the Thomson ring itself is a little bit tricky to calculate. One my search in engineering books as Grover [9] or Philippow for inductance calculations of a cylindric ring [Fig. 6]. Whenever you find such a formula it´s full with strange numbers and quotients. Mostly it´s not explained how the formula is derivated. I myself do not trust such formulas without knowing the background. It´s to risky that all the simulations went wrong just because a failure in an old 100 years old formula. So the best way in my opinion is to use formula [3.13], which is absolutly derivated analytically and see how to deal with that...
Imagine a cylindric ring (Fig. 6) filled with many small filament ring circles (number n) inside. Each of the ring circles carries a current Ii=I/n. The distance between all the filament rings is constant. Then the total energy of the system is:

total energy inductance system [4.1]
Assuming an averaged current in the filaments we may write:
Energy Averaged Current in Filaments [4.2]
Elimination of the current I gives the total inductance of the Thomson ring
Inductance Thomson Ring[4.3]
Via numerical calculations of equation [3.11] one may calculate all elements of the inductance matrix of the Thomson ring. Before the filament density has to be defined. I decided to use the "unit filment per mm".
Filament Density. [4.4]
For a well defined filament density eta the radial number for the filaments is
radial filament density . [4.5]
Analogical the vertical number is
vertical filament density [4.6]
The total number of filaments is
total number filaments[4.7]

  Modeling of a Thomson Ring - Inductance Calculation of a Ring with rectangle cross section

Fig. 6: Modeling of the Thomson Ring with ring filaments inside the ring. The section of the ring is splitted up into nrad radial filaments and nh vertical filaments. The radius of each filament wire is so defined that each filament surface touches the surface of the neighbour filament. The partial inner inductance is discussed in equation [4.9].

For a endless round wire the partial inner inductance (watch also equation [3.9]) is
partial inner inductance endless round wire [4.8]
This inner inductance is independent of the radius r of the wire. The inner inductance of a wire with square section (width a, height a) is a all little bit smaller than the inner inductance of a endless round wire:
Energy InductancePartial Inner Inductance square conductor[4.9]
Via precise numerical field integration Iwas able to show that the inner inductance of a endles wire with square section is 48,3196 µH/m, which is around 96,6% of the partial inner inductance of a endless round wire with dL/dl=50nH/m. However for simulations I use
appr. inner Inductance square section wire.[4.10]

4.2 Resistance of the ring

In 3.3 we approximated the wire resistance with equation [3.21], which is acceptable for the coil wire. In the Thomson ring experiment cylindric "Thomson-rings" were commonly used. According to the winding ratio of the coil to the "Thomson Ring" (one winding) and the coupling, the current flow inside the Thomson ring is mostly around ten times larger than in the coil. So here it´s better to calculate the resistance more serious.
The conductance of a wire with homogenous current density is
so, the differential expression depending on the radius is
Differential Conductance Cylindric Ring[4.12]
with the substitution dA=hdr and l(r)=2*Pi*r we get by integration over the radius r
Conductance Cylindric Ring. [4.13]
As the resistance is the inverse of the conductance, we may write
Resistance Cylindric Ring[4.14]
which is a good formula to calculate the ohmich resistance of a cylindric ring. Depending on the discharge frequency of the circuit we may also regard the skin deep of the current density in the Thomson ring.
  Thomson Ring Model

Fig. 7: Thomson Ring as a cylindric ring. r1 is the inner radius, r2 is the outer radius. d ist the height of the ring. The resistance of the coil has to be calculated by integration (equation [4.13]) to regard the different circumferences. At higher frequencies the distribution of the current density is not homogen. Then it´s usefull to correct the calculated resistance from equation [4.14] due to the skin effect.


Baustelleskin effect calculations....

5. Mutual Inductance between Coil and Ring

5.1 Mutual Inductance via the Mutual Inductance Matrix

For simulations we need the value of the mutual inductance matrix for all possible heights of the Thomson ring while the acceleration. The inductance matrix of a system with n-1 circles in a flat coil and one circle with index n (Thomson ring) above the coil is:

Mutual Inductance Matrix[5.1]
The mutual inductance between the coil and the Thomson ring is the sum of all matrix elements of the right column (blue deposited) except the last element Ln,n, which is the self inductance of the Thomson ring. The summ of all other matrix elements except the last row is the total inductance of the coil (yellow deposited).
In reality one often uses more than one layer in the coil. Taking this into account we have to take the the right column except the last element of all inductane matrices from all layers in the coil:
Total Mutual Inductance Matrix of a Coil[5.2]
This is the mutual inductance matrix for a system containing j layers of flat coils stacked about each other to one large coil and a Thomson ring with inductance Lnn above the coil in a certain distance.

  Mutual Inductance Components
  Fig. 8: Mutual inductance components between a flat coil and a Thomson ring above the coil. The windings of the coil have got the indices 1,2,3,..., n-1. The current throw any coil circle is equal. The Thomson ring is placed above the coil with index n. The mutual inductance between the coil and the ring may be calculated with the inductance matrix shown on the left.


  Mutual Inductance of a multi layer coil

Fig. 9: Mutual inductance components between a flat coil with j layers and a Thomson ring above the coil. The windings of each layer of the coil have got the indices 1,2,3,..., n-1. The current throw any coil circle is equal. The Thomson ring is placed above the coil with index n. The mutual inductance between the coil and the ring may be calculated with the inductance matrix shown on the left.

5.2 Fitting with Multi Exponential Functions

While the accleration of the Thomson ring away from the coil the mutual inductance between the coil and the ring changes. Imagine a height spectrum for the ring positions y=[yo, ..., ye]. At the beginning the ring has got a height of yo above the top laywer of the coil (layer I). While moving upwards the mutual inductance M(y) decreases a lot. One may write a programm to simulate all this behavior by calculating only the values M(yi) you need for certaion steps i in your recursive algorithms. I myself prefer a moore analytical proceed: After defining the simulation intervall y=[yo, ..., ye] I spilt it up into 1000 parts with aquidistant intervall lenghts, so that we get an array of mutual inductance values: M1(y1), M2(y2), M3(y3),.... M1000(y1000). My standart flat coil FL1 has got 14 windings, a inner radius r1=5mm and a winding distance of 5mm between each ring. Lets suppose a Thomson ring with radius rring=52mm. The values M(y) may be calculated numerically by using equation [3.11] and [5.2]. Afterwards the gained data may be fitted well with a double exponential function [5.3]:
Double Exponential Fit Mutual Inductance
In the case of the FL1-coil and the Thomson ring with rring=52mm I got the following fit parameters:
Double Exponential Fit

Mutual Inductance Fit Function


Fig. 10: Mutual Inductance between the flat coil FL1 and a Thomson Ring with r=52mm. The distance between the coil an the ring is [1...20cm]. The values for the mutual inductance where calculated numerically by using equation [5.2] and [3.11]. The simulated values M(y) are fitted with a double exponential function [5.3]. The relative failure of the fit function compared to the original data is max. around 1,5E-3. For small distances the failure is even less.

As you see in Fig. 10 double exponential functions are great for fitting the mutual inductance functions M(y). Another advantage is that´s quite easy to claculate the derivation dM(y)/dy. The derivation is quite important for calculate the accelertion force of the ring, we will handle in the next chapter.

6. Calculation of the acceleration force

6.1 Via the magnetic field

By integration of the maxwell equation we get:maxwell equation, Stokes[6.1]
The induction voltage is the time derivation of the total flux throw the ring.
induction voltage [6.2]
Flux Phi12 (Fig. 11) may be calculated by surface integration of the flux density Bz
Flux Phi 12 with surface integration [6.3]
Phi 2 is the flux created by the current I2(t) in the ring. Integration gives:
Flux Pgi 2 with surface integration [6.4]
By this way one has to calculted first the Bz-components of the magnetic field created by the coil and the ring via Biot Savart law. Afterwards one has to integrate the magnetic field density (equation [6.3], [6.4]) numerically. Fortunately it´s easier to use the correlation between flux, inductance and currents
linked flux[6.5]

Eddy current induction rings


Fig. 11: Eddy current induction: A time dependent current through Coil L1 creates a magnetic field, which may be calculted by Biot Savart law. A certain part of the total flux Phi12 flushes throw the Thomson ring above the coil. This flux may be calculated via surface integration of the magentic flux density in the ring plane (equation [6.3]) or just by using the mutual inductance expression (equation 6.5).

Using equation [6.2] we may write directly
induction voltage [6.6]
The mutual inductance L12=M varies by time, so that we get the expression induction voltage expression [6.7]
With Ohm´s law we get for the current
current in Thomson Ring[6.8]
The Lorentz force Fz(t) created by the current I2(t) then is
Lorentz Force Thomson Ring.[6.9]
As shown in Fig. 12 only the Bz-components in the ring plain are responsible for the induced current I2.

Lorentz Force at the Thomson Ring Experiment


Fig. 12: Lorentz force Fz: The Lorentz force on the Thomson ring depends on the radial magnetic field Br inside the ring and the current flow inside the ring I2. The current flow I2 depends on the changing of the Bz- components of the magnetic field created by the coil.

The Br-components inside the ring are responsible for the Lorentz force (equation [6.9]). So, finally the lorentz force depends on a very tricky relation of the Br- and Bz-components.

6.2 Via Gradient of mutual inductance

Another moore practical method to calculate the acceleration force is to use the gradient of the mutual inductance. The total magnetic energy of the system is:

Total Inductance Energy Thomson Ring Experiment[6.10]
Mechanical variation of the magnetic energy gives the differential expression
variation - differntial energy expression. [6.11]
Differentiation to the height y yields:
energy differential to height [6.12]
The corresponding force to the potential E is described by the gradient [1], [2]:
Force as Gradient of Mutual Inductance[6.13]
Therefore the acceleration force in z-direction is:
Acceleration Force Thomson Ring [6.14]
With the double exponential function (equation [5.2]) we may easily write the expression
Derivation Double Exponential Fit Mutual Inductance. [6.15]

Gradient of Mutual Inductance - Force Calcultion


Fig. 13: Coherence between the gradient of mutual Inductance and force: Fz is proportional to the currents I1, I2 and the gradient of mutual inductance between the ring and the coil. With this method there is no need to calculate the magnetic field components Bz, integrate over the flux density and calculte Br.

which is very usefull to calculate the acceleration force (Fig. 13).

6.3 Table of Differential Equations

After discussing all the inductance theory and force calculations we are now ready to deduce all differential equations needed for numerical simulations.
With Kirchhoff´s voltage law we may write for the first circuit:
Differential Equation1[6.16]

Neglecting the line inductance Lline we may set L1=Lcoil (remember Fig. 1). Then the the Voltage UL1(t) is equal to the changing of the magnetic flux through the circuit:
Differential Equation 2[6.17]
The magnetic flux through the first circuit is the flux created by the inductance L1 and the flux created by the inductance L2. So we may write
Differential Equation3[6.18]
Differentiation gives
Differential Equation 4, [6.19]
while dL1/dt=0. Solving the equation to dI1/dt gives:
Differential Equation 7[6.20]
With Kirchhoff's second law the second circuit gives:
Differential Equation 8[6.21]
Differentiation to time yields
Differential Equation 9.[6.22]
Differential Equation 10[6.23]
So finally we get:
Differential Equation 11[6.24]
Solving to dI2(t)/dt gives:
Differential Equation 12[6.25]
Putting dI1(t)/dt into dI2(t)/dt gives:
Differential Equation 13[6.26]
After solving dI2(t)/dt we get:
Differential Equation 14 [6.27]
For dI1(t)/dt we get:
Differential Equation 15 [6.28]

The Voltage on the capacitor Uc(t) may be described by
Differentiation to time gives:
Voltage Differential[6.29]
Voltage Function
Now remember the expression for the mutual inductance between coil and ring (Chapter 5.2, equation [5.3]):

Double Exponential Fit Mutual Inductance[6.30]
In chapter 6.2 we found:
Derivation Double Exponential Fit Mutual Inductance[6.31]
Differentiation to time of equation [6.30] yields:
Time Derivation Double Exponential Fit Mutual Inductance [6.32], which allows us to describe dM(y)/dt with the speed function v(t)=dy/dt of the Thomson ring. With newtons second law we get the accelleration of the ring:
Newton 2.Law [6.33]
Electromagnetic Acceleration [6.34]
with the force F(y(t)) we deduced in chapter 6.2:
Force as Gradient of Mutual Inductance[6.35]


Maybe you miss a minus, because the force is a negative Gradient. Since the currents I1(t) and I2(t) are rotating in opposite directions (watch Fig. 17 later) and our quotation in equ. [6.34 ] the product I1(t)*I2(t) is negative for 0<t<1/3Pi*sqrt(LC), which is the main accelleration time intervall. With dM(y(t))/dt<0 for every time we get a positive force in y-direction with expression [6.35]. Substitution of dM(y(t))/dt gives the formula for the acceleration of the ring in y-direction:
Acceleration Formula Thomson Ring Experiment [6.36]
Now we may put all found expressions in one table of differential equations:

Nr. Boundary
Differential Equation solved function
0 I1(0)=0 Differential Equation Current Capacitor Bank I1(t)
1 I2(0)=0 Differntial Equation System 2 I2(t)
2 Uc(0)=Uo Capacitor Voltage Differnetial Equation Thomson Ring Experiment Uc(t)
3 y(0)=0 Speed of the ring y(t)
4 v(0)=0 Acceleration Speed Differnetial Equation Thomson Ring Experiment v(t)
5 E1(0)=0 Absorbed Energy CLR Circle E1(t)
6 E2(0)=0 Absorbed Energy Thomson Ring E2(t)
7 A(0)=0 Action Integral CLR Circle A(t)

Before solving the differential equations the functions M(y(t)) and dM(y(t))/dt must be substituted with:

Double Exponential Fit Mutual InductanceTime Derivation Double Exponential Fit Mutual Inductance

These 8 differential equations may be solved numerical using Runge Kutta Algorithms. I prefer using adaptive step size regulation methods as the Cash Karp algorithm. By using this method I built a graphical used interface programm, which is great for realistic simulations of the Thomson ring experiment.

7. Simulation of realistic experiments
7.1 Software

The simulation software I built contains a front panel where all relevant data (Capacitor Bank Data, Spiral Coil Data, Algorithm Parameters) may be putted in (Fig. 14). After putting in all needed data the inductance of the Thomson ring is calculated first by using the method described in chapter 4.1.

Simulation Software Thomson Ring Experiment
Fig. 14: Front Panel of the Simulation Software: In the top all needed data may be entered. The Thomson ring inductance is approximated by n ring filaments depending on the Filament-density.

In a next step the programm calculates the mutual inductance between the Thomson ring and the coil by using the method from chapter 5. The mutual inductance curve is fitted by a double exponential function automatically (Fig. 15).

Simulation Software Thomson Ring- Gauß GunFig. 15: Mutual Inductance Fit: Fit tolerance and mumber of iterations may be adapt to the dimensions of a specific experimental setup. On the right side a work coil (here one layer) and the Thomson ring above is shown.

The inductance matrix of the work coil is calculted by the methods presented in chapter 3.2. It´s possible to define a work coil with several layers for higher inductances. Summation over all ring resistances of the work coil gives the total resistance of the coil (chapter 4.2). All these data are used in the software to substitute the parameters of the differential equations. Numerical solving of the differential equation system via Runge Kutta Algorithms enables the software to show many interesting graphs (Fig. 16):

Voltage, Current Graphs Thomson Ring, Induction Gauß Gun, Coil GunFig. 16: Simulation results 1: a) voltage and current at the capacitor b) Sum of all energies c) Height force Graph d) position height of the ring e) speed of the ring f) kinetic energy of the ring

The discharge current of the electrical circle may be compared with the induced current in the ring in one graph (Fig. 17a). The Action integral value may be analysed as a function of the time (Fig. 17b) to reduce over current damaging effects of the used capacitor.

Current Curve, Action Integral
Fig. 17: Simulation results 2: a) current in the ring (plot 0) and discharge current (plot1) b) Action Integral curve

Futher the magnetic energies (Fig. 18a), capacitor energy (Fig. 18b), mutual inductance energy (Fig. 18c) are plotted. Together with the total heat energy in the discharge circle (Fig. 18d), the heat energy in the Thomson ring (Fig. 18e), the sum of all energies (remember equation [1.9]) of the system is presented (Fig. 18f). With this graph the energy deviation from the start energy may be analysed to get a feeling how accure the simulation works. In the presented example the maximal deviation is only appr. 0,6mJ, which is really small compared to the total energy of 1200J.

Magnetic Energy, Energy in Capacitor, Energy in Mutual Inductance, Heat Energy, Sum of all Energies

Fig. 18: Simulation results 3: a) magnetic energy of inductance L1, L2 b) capacitor energy c) mutual inductance energy d) heat energy e) heat energy in the ring f) sum of all energies

Comparing al the plotted energies one may easily find some parameters wich may be improved to get a higher efficiency. In the next chapter I want to compare three different flat coils to find out the specific differences.


7.2 KB1-KB2-FL1,2,3 Experiment Simulations

While the project Metallverformung durch Kurzpulsmagnetfelder 2006 I developed a set of different flat coils with an inductance of 11uH, 45uH and 100uH. Assuming a Aluminium ring with r1=50mm, r2=53mm, h=6mm one may simulate a pulse discharge of the capacitor bank KB1-KB2 to accelerate the Thomson Ring. Calculation of the standart resonance frequency and taking the skin effect into acount the following table of data is obtained:

used Coil
measured coil inductance
calculated coil inductance
standart frequency f0
Thomson ring resistance correction
3240 Hz
1890 Hz
1330 Hz

Due to the skin effect I used a correction of the electrical resistance depending on the standart resonance frequency. By simulation of these experimental setups I got the following results:

Height of the Thomson Ring, Gauß Gun Simulation Speed Simulation of the Thomson Ring Experiment, Induction Gauss Gun, Coil Gun
Fig. 19a: Height of the Thomson ring: With the FL1-Setup the acceleration of the ring is the highest within the first 100us. Anyway with the FL2-Setup one gets the fastest gain of height. Fig. 19b: Speed of the Thomson ring: The final speed of the ring with FL2 is 129,2 m/s. With the FL3-Setup the maximal speed is 109,8m/s compared to the FL1-Setup with 109,8m/s. The end speed is achieved at around 1ms.

The acceleration of the Thomson ring with the FL1-Setup is the highest in the first 100us (fig. 19a). After 200us the speed of the ring in the FL1 experiment is much less than in the other experiments (Fig. 19b). After 1ms there is no moore significant acceleration. It´s obvious that the highest ring speed may be achieved with the FL2 experiment. In my old experiments I also got the best electromagnetic forming results by using the FL2-coil. The discharge current flowing throw the coils has got a absolute maximum of appr. 12,8kA in the Fl1 experiment, compared to 7,1kA with the FL2 and around 5kA with the FL3 experiment (Fig. 20a). The induced current in the ring is much higher than the discharge current flowing throw the coils (Fig. 20b). Surprisingly the maxima of the ring currents are quite similar between 40kA´s and 45kA´s.

Discharge Current Pulsed Power Acceleration Eddy Current Pulsed Power Acceleration Thomson Ring
Fig. 20a: Discharge current of the circuit. Fig. 20b: Induced current in the Thomson ring: The current maximum in the ring is very similar for the FL1 and FL2-Setup with around 45kA.

Capacitor voltage curves are important because the reversal voltage is the main damaging factor reducing the life time of high voltage capacitors. In the simulations the reversal voltage is between -3,0kV and -3,5kV (Fig. 21a). This indicates a low damping of the circuit enery oszillation. Numerical solving of the the differential equations number 5 and 6 (table in chapter 6.3) the absorbed heat energy in the circuit (R1) and the ring (R2) may be calculted. The results are plotted in Fig 21b. The heat energy in the ring is quite similar for all used coils (dotted lines in Fig 21b). This means that the numerical value of the action integral A(t) of the ring current is similar for any used coil. The absorbed heat energy in the circuit is mainly affected by the Action Integral, whose value compensates the small resistance of the FL1 coil compared to the other coils. This strange effect yields to a fast heating of the FL1 coil (blue line FL1-heat R1 in Fig. 21b), although this coil has got the lowest restistance. The heat absorbtion curves of the FL2 and FL3 coils are significant less rising.

Pulsed Power Discharge Capacitor in Thomson Ring Experiment Heat Energy in Thomson Ring
Fig. 21a: Voltage curves Fig. 21b: Heat energies
Nevertheless for long times the absorbed energy in the circuit is quite similar for all investigated experiments. The absorbed circuit heat energy in the FL1-experiment is 80,1% of the start energy, compared to 76,6% in the FL3 and 76,2% in the FL2 experiment (Fig. 22).

The total ring heat energy is around 12-13% in the experiments. With the FL2 experiment one may get the best mechanic efficieny, so that the kinetic energy of the ring reaches 10,9% of the start energy. The kinetic energy of the Thomson ring in the FL3 experiment is 10,7% followed by 7,9% in the FL1 experiment.

So the efficiency of acceleration is around 10% of the start energy in in the capacitor for all investigated experiments using the coils FL1, FL2 and FL3.
Energy Distribution Thomson Ring Experiment
  Fig. 22: Final Energy distribution of the Thomson ring experiment

8. Summary and Discussion

In this Project the Thomson ring experiment was analyzed analytically. Ensuing from a circuit diagramm (Fig. 1) all relevant parameters as resistances, inductances and mutual inductances were regarded. The theoretical background of inductance was deduced from the Poisson equation in a fairly detailled manner. Concepts as inductance matrix and mutual inductance matrix were conceived to calculate the inductance of an arbitrary flat coil with multi layers. The inductance of the Thomson ring was approximated by placing multi filament rings inside the Thomson ring. Due the mutual inductance between the coil and Thomson ring is a variable function depending on the coil-ring-distance I tried to find a mathematical expression which describes this relation as accurately as possible. Comparison of different fit functions resulted in an excellent fit capabilitiy with double exponential functions. By this method all coupled differential equations could deduced analytically without the need of indroducing expressions as elliptic integrals inside the differential equations. Furthermoore the time differentiation of mutual inductance dM(y)/dt could described easily by writing dM(y)/dt=dM(y)/dy*dy/dt and replacing dM(y)/dy with the trivial differential expression of the founded double exponential function. The acceleration force calculation was made possible without using the lorentz force expression inculding radial magnetic flux density calculations, which would have required using Biot Savart law inside the differential equations. Instead of this awkward procedure the gradient of mutual inductance could be used as an easy alternative. The founded differential equation system could be solved numerically by using Runge Kutta Algorithms with step size regulation.

With a self built software the evaluation of simualtion results gives now the opportunity to improve practical experiments. The suitability of the built software could be tested by the simulation of the Thomson ring experiment with my coils FL1, FL2 and FL3 and the capacitor bank KB1-KB2 as energy source. Analysis of the simulations gave a maximum acceleration efficiency of around 10% with the used Thomson ring. The graphs of ring position, speed, voltage, currents, heat energies may be obtained and analyzed to understand the differences.
In further practical experiments these results should be validated by measurements. This could lead to a even better adjustment of the simulation parameters as skin effect corrections and sparc gap resistance influence. In a next project I will analyse the eddy current density and pressure distribution of larger metal sheets instead of using a Thomson ring....


9. Literature

[1] Introduction to Electrodynamics, Addison Wesley; 3rd edition, David J. Griffiths
[2] Classical Electrodynamics, John David Jackson
[3] Simulation and verification of Thomson actuator systems, Ara Bissal, Göran Engdahl, Ener Salinas, Magnus Öhström, COMSOL Conference Paris 2010
[4] HIGH-SPEED AC CIRCUIT BREAKER AND HIGH-SPEED OCR, Young-woo JEONG, Hyun-wook LEE, 22nd International Conference on Electricity Distribution
[5] Shape Optimization of a Thomson Coil Actuator for Fast Response Using Topology Modification, Wei Li and Chang Seop Koh, Journal of Electrical Engineering & Technology Vol. 7, No. 1, pp. 58~63, 2012

[6] Optimizing Thomson’s jumping ring, Paul J. H. Tjossem and Elizabeth C. Brost, Department of Physics, Grinnell College, Grinnell, Iowa 50112
[7] Klassische Mechnik, F. Kuypers, Wiley-VCH, 5.überarbeitete Auflage
[8] Formulas and Tables for the Calculation of Mutual and Self Inductance, E. B. Rosa, F.W. Grover, U.S. Dept. of Commerce and Labor, Bureau of Standards : 1912
Inductance Calculations, F.W. Grover, Dover Books on Electrical Engineering, 1 edition, 2009
[10] Metallverformung durch Kurzpulsmagnetfelder, Max Bigelmayr, Jugend forscht Projekt 2006
[11] Discharge behavior of capacitor banks, Max Bigelmayr, 2011
Berechnung der Magnetfelder von Spulen, Max Bigelmayr, 2010

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May 2015, August 2015 (V1)
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