Regular paper - esrgroups.orgjournal.esrgroups.org/jes/papers/JESSI23.pdfThe self inductances L ss and L aa are constant during the operation of the launcher unlike the mutual inductance

Corresponding author : [email protected] Laboratoire des Systèmes Eléctromagnétiques, EMP Bordj El-Bahri, Algiers 16111, Algeria. Copyright © JES 2010 on-line : journals/esrgroups.org/jes

M. Bensalem Y. Ouazir

J. Electrical Systems: special issue N° 2 (2010): 26-36

Regular paper

Analysis of Linear Induction

Launcher Performance Fed by Capacitor Banks Based on

Coupled Circuits Model

This paper is concerned with modeling of linear induction launcher fed by capacitor banks taking their transient behavior into account. The modeling off the both electrical and mechanical transient phenomenon under capacitor banks excitation is performed and simulation of relative movement between induced and fed parts is done, a computer simulation results are presented.

Keywords: Electromagnetic Propulsion, Linear Induction Launcher, Coupled Circuits, Capacitor Banks, Movement.

1. INTRODUCTION

Linear induction launcher, have received increased attention because they have important advantages over other electromagnetic launchers. The first is the absence of contact between the moving and stationary parts, thus permitting repeated use of the barrel. But the second concerns the distribution of the stresses over large surface areas, thus permitting the use of heavier projectiles.

The LIL consists to two principal parts: the barrel coils, energized by capacitor banks, and the projectile enclosed in a cylindrical conducting sleeve. An accelerating Lorenz forces are provided by the interaction between the magnetic wave produced by the barrel currents and the currents induced in the projectile sleeve. The barrel is usually divided into many sections each of which is operated at a different constant frequency from an independent power supply [1], [2].

This paper presents a coupled circuit method used to computing the both electrical and mechanical transient performances of the linear induction launcher (LIL) powered by capacitor banks as indicated in the diagram fig.1. In this method, the self and mutual inductances of electrical circuit model of LIL are calculated analytically. A computer simulation results based on this model are presented.

J. Electrical Systems SI 2 (2010):: 26-36

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2. BASIC MODEL

The stator winding of the LIL consists of a coaxial arrangement of solenoidal coils, which are appropriately connected in series and excited from a sinusoidal supply. The armature consists in a simple cylinder of conducting material. The projectile and barrel have the same axes and the currents induced in the projectile flow in circumferential paths.

In the coupled circuit method the projectile may be divided into an equivalent set of conducting rings, in each of which the current density is approximately uniform. Each ring may now be regarded as a circuit, in much the same way that the barrel windings form circuits. The projectile circuits have not externally-impressed voltage Know. [3]

Fig.1. Schematic diagram of multi-stage capacitor driven LIL.

Fig.2. circuit model of the capacitively driven coil launcher.

If the projectile is free to move in the z-direction, using Kirchhoff’s voltage law shown in fig.2 the equations governing the electrical behaviour of the launcher may be written in the form [3]:

.s a s azs s s s s s a a

d I d I d Md SV R I L M Id t d t d t d t⎛ ⎞⎜ ⎟⎝ ⎠

= + + +

M. Bensalem et al: Analysis of Linear Induction Launcher Performance Fed...

28

0 . a sa sa a a a a s s

d I d I d S z d MR I L M Id t d t d t d t

⎛ ⎞⎜ ⎟⎝ ⎠

= + + +

(1)

The magnetic energy stored in the system is given as

1 12 22 2

E L I L I M I Ls s r r sr s r= + + (2)

Note that the variable M represents the mutual inductance of the coil pair is a function of z. The force Fz on the armature is therefore given as:

Z a sdMdEF I I

dZ dZ== (3)

The acceleration expression is:

a sZZ

I IF dMam m dZ

== (4)

The self inductances Lss and Laa are constant during the operation of the launcher unlike the mutual inductance M and the inductance gradient G which are functions of elliptic integrals and are readily calculated as functions of stator and armature coil geometry and separation.

3. INDUCTANCE CALCULATIONS

The self and mutual inductances that are the elements of the matrices Lss, Lsa and Las are calculated, using an analytical expression of the mutual inductance between two coaxial circular filaments (Fig. 2). This expression, for two coil filaments separated a distance z is given by the relation derived by Silvester [4], [5]:

( ) ])(2)(12),,( 220 kEk

kKk

krrzrrm jiji −−⎢⎣⎡= μ (5)

( )2 2

4 i j

i j

r rK

r r Z+=

+ (6)

Where K(k) and E(k) are complete elliptic integrals of the first and second kind. The partial derivative of m, is readily taken to determine the gradient:


29

}{ 2 202 2(1 ) ( ) (2 ) ( )

4(1 )ij

i j

m k z k K k k E kz k r rμ∂

= − − −∂ −

(7)

Fig.2. Diagram for filamentary inductance formula

3.1 Mutual Inductance and Mutual Inductance Gradient

Working with the general arrangement shown in Fig.3, the cross sectional area of the coil and projectile is divided respectively to Nsxns and Naxna elements. The mutual inductance between an ith element in the stator and the jth element in the armature is given by:

jas

assa im

nnNNM =Δ (8)

The mutual inductance between the stator and the armature is given by the summation of all mutual inductances between all pairs of filamentary coils:

∑ ∑∑ ∑ =Δ== == =

s

i

a

j jas

assn

i

an

jsasa

n nim

nnNNMM

1 11 1 (9)

Similarly the inductance gradient is given as:

1 1

s nn a ijas a

i j

ss a

mN NGn n z= =

∂= ∑ ∑ ∂

(10)

3.2 Self Inductances

Continuing with the general arrangement shown in fig.3 the self inductance of each coil is calculated using Grover’s expression for the inductance of a


30

single turn circular coil of square cross section given below for the stator coil, calculation of the armature coil self inductance is done in similar manner.

2 1

1 1 12

n n ns

ij ii j i is

ssNL m ln

−

= = + =

⎡ ⎤⎛ ⎞⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

= +∑ ∑ ∑ (11)

According to Grover [5] the term li is given by:

0 81 ln 0.2041 0.84834 2 6

ii i i

il ξμ ξ ξ

ξ⎡ ⎤⎛ ⎞⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

= + + − (12)

With: 2

ii

i

ca

ξ ⎛ ⎞⎜ ⎟⎝ ⎠

=

where ci and ai are respectively the mean diameter and side length of ith

element.

Fig.3.division of real coils into an array of filamentary coils The equations (9) to (12) can be used to calculate the components of the matrices Gsr , Lss , Lsr and Lrr in a systematic manner.

4. TIME STEP NUMERICAL SOLUTION

The differential equations (1) arranged into a matrix form allow for convenient expression and manipulation when solving the time step numerical problem is shown below:

[ ] [ ] [ ] [ ] [ ] [ ] [ ]. .

. . . . .V L I M I R I v G I⎡ ⎤ ⎡ ⎤= − + −⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ (13)

In which:


31

0sV

V⎡ ⎤⎢ ⎥⎡ ⎤⎣ ⎦ ⎢ ⎥⎣ ⎦

= Coil voltage Matrix (14)

[ ]0

0

ss

aaL

L

L

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= Self Inductance Matrix (15)

[ ]0

0

sa

as

MM

M

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= Mutual inductance Matrix (16)

[ ]0

0

sa

as

dMdz

GdM

dz

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= Mutual Inductance Gradient Matrix (17)

[ ]0

0ss

ss

RR

R

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= Resistance Matrix (18)

[ ]a

sI

I

I

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= Coil Current Matrix (19)

..

.a

sI II

⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= Rate of Coil Current Change Matrix (20)

The relation between the capacitor and driven coil currents are

[ ] c adC V Idt

= −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ (21)

Where [ ]C is a diagonal containing the capacitances and cV⎡ ⎤⎣ ⎦ and aI⎡ ⎤⎣ ⎦are the

capacitor voltage and the drive coil current matrix respectively. cV⎡ ⎤⎣ ⎦ and

aI⎡ ⎤⎣ ⎦are submatrices of [ ]V and [ ]I . Placing all the differential terms on the left hand side and solving for the differential current matrix we find:

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ].

. . .zL M I V R I I v G I⎡ ⎤⎡ ⎤ = − − −⎢ ⎥⎣ ⎦ ⎣ ⎦ (22)


32

[ ] [ ] [ ] [ ][ ] [ ] [ ]. 1

. . . .zI L M V R I v G I−⎡ ⎤ ⎡ ⎤ ⎡ ⎤= − −⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦

(23)

Using a subscript notation for a time step procedure

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]1

.1 . . . .t t ztt t t t tI L M V R I v G I

−

+⎡ ⎤ ⎡ ⎤+ − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= (24)

With the initial condition given by setting

[ ]0 0I = and 0 0initialV

V⎡ ⎤⎢ ⎥⎡ ⎤⎣ ⎦ ⎢ ⎥⎣ ⎦

= and vz0=0

[ ] [ ] [ ]1

0 0 00

..I L M V

−⎡ ⎤ ⎡ ⎤= −⎢ ⎥ ⎣ ⎦⎣ ⎦ (25)

Thus 010 0

. .. .I I t I t I⎡ ⎤ ⎡ ⎤

=⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

= +Δ Δ (26)

The solution for the current matrix generally given as:

1

..t tt

I I t I+

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎣ ⎦= +Δ (27)

The force acting on the armature (and thus the acceleration) is greatly dependant of the product of the calculated currents. It is given at time t by:

1 11 1. .T

z t s sr t rt tF I G I+ ++ +

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦= (28)

In the movement part, the acceleration is obtained directly by dividing the obtained force by the projectile mass m:

( 1)( 1)

z tt

Fm

a ++ = (29)

The projectile velocity Vz and displacement Sz are then determined through a numerical integration of acceleration and velocity respectively. The accuracy of this solution is obviously governed by the magnitude of the time step �t used and the positional accuracy to which the mutual inductance [M] and its gradient [G] matrices are solved. They are given by:

( 1) ( 1).z t t z tV tV a+ += + Δ (30)

( 1) ( 1).t t z tS tS V+ += +Δ (31)


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5. APPLICATION

The prototype tested in this application is constructing in Electromagnetic Systems Laboratory (EMP). It consist a 100g Aluminum projectile. The simulation model parameters are given in Table 1.

This study assumed that the starting section of the LIL consists of 6 drive coils connected in a three capacitor banks with phase sequence A,-C,B, -A,C,-B each with capacitance 1500µF . The projectile is assumed to be segmented in to 20 coaxial rings. The dimensions given in table 1 are used in simulation, capacitors were initially charged up to 500v, 450v, 400v in each phase respectively. The three phase capacitor banks were fired in sequence: first phase A, then phase C after 60o with negative initial capacitor voltage, and last phase B after another 60o, the fig.4 shows the three phase capacitor discharge currents. It is seen that these currents are strongly attenuated. Because the projectile dissipates a lot of energy. The velocity and force profiles are smooth, as shown in Figures (5) and (6).

The mutual inductances among all of the drive and projectile coils are calculated during each incremental movement of the projectile. The method provides results for the coil and a projectile ring mutual inductance; coil phase current, velocity and mechanical forces acting on the projectile. Fig.7 and Fig.8 shows the mutual inductance between the coil in the starting section and projectile ring and there gradient dM/dz. When the centre of the coil and the ring are aligned, there is no axial force because the change in mutual inductance is zero.

TABLE I DIMENSION FOR SIMULATION MODEL

Name Value Unit Starting Section : Length 200 mm Number of coils 6 - Number of turns per coil 315 spires Axial length of coil 30 mm Outside diameter 48 mm Inside diameter 21 mm Sleeve dimensions : Length 200 mm Outside diameter 17.5 mm Inside diameter 15.5 mm Material Aluminum - weight 100 g


34

0 5 10 15 20 25 30 35 40 45-300

-200

-100

0

100

200

300

400

500

600

curr

ents

(A

)

Time(ms)

Fig.4. Three phase currents in the starting section

0 0.05 0.1 0.15 0.2 0.250

1

2

3

4

5

6

pos(m)

vit(m

/s)

Fig.5. Velocity profile of generator-driven in starting Section


35

0 0.01 0.02 0.03 0.04 0.05 0.06-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

For

ce (

kN)

Time (sec)

Fig.6. Force

0 0.05 0.1 0.15 0.2 0.25-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

-5

Mut

ual i

nduc

tanc

e(x1

0H)(

H/m

)

Displacement (m)

Fig.7 Mutual Inductance


36

0 0.05 0.1 0.15 0.2 0.25-1.5

-1

-0.5

0

0.5

1

1.5x 10

-4

Mut

ual I

nduc

tanc

eGra

dien

t(H

/m)

Diceplacement (m) Fig.8.Mutual Inductance Gradient

6. CONCLUSION

The analytical study for linear induction launcher based primarily on the inductances calculation has being capable to model transient and dynamic states. The capacitor banks discharge derive the alternating current needed to create a traveling wave from resonance with the inductance of the coils.

REFERENCES

[1] J. L. He, Z. Zabar, E. Levi and L. Birenbaum, “Transient Performance of Linear Induction Launchers Fed by Generators and by Capacitor Banks”, IEEE Transactions on Magnetics, vol. 27, no. 1, Jan 1991.

[2] B. Azzerboni. E. Cardelli , M. Raugi , A. Tellini, “Some Remarks on the Current Filament Modeling of Electromagnetic Launchers”, IEEE Transactions on Magnetics, vol. 29, no. 1, Jan 1993.

[3] J. L. He, E. Levi, Z. Zabar and L. Birenbaum, “Concerning the Design of Capacitively Driven Induction Coil Guns,” IEEE Trans. on Plasma Science, Vol. 17, No. 3, pp. 429-438, June 1989.

[4] P. SILVESTER, “Modern Electromagnetic Fields”, Prentice-Hall, Englewood Cliffs, NJ. 1968.

[5] F.W. GROVER, “Inductance Calculations”, Dover Publications, New York, 1946.

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Regular paper - esrgroups.orgjournal.esrgroups.org/jes/papers/JESSI23.pdfThe self inductances L ss and L aa are constant during the operation of the launcher unlike the mutual inductance