Research on the Scaling Model of Electromagnetic Coil Launcher

Zou Bengui, Li Ruifeng, Chen Xuehui, and Cao Yanjie

Naval Aeronautical and Astronautical University, Yantai, Shandong 264001, China

Abstract—Model experiment is an effective method to investigate the working characteristics of electromagnetic coil launcher (EMCL). Results from the model experiment are applied to the prototype study of EMCL after the scaling relations of physical quantities between model and prototype are understood. A new type of EMCL is designed and the scaling factor relations of the physical quantities between model and prototype in EMCL are deduced based on its mathematical model and similarity theory. Its scaling model is investigated by simulation. It is shown that the scaling factor relations of the physical quantities are expressed as length scaling factor and the scaling model is tenable. Index Terms—Electromagnetic coil launcher, prototype, model, length scaling factor, model experiment

I. INTRODUCTION Electromagnetic coil launcher (EMCL) is an important branch

of the electric launcher family with many valuable merits and properties such as no mechanical contact between projectile and barrel, reasonable mechanical structure, higher launch efficiency and capability of launching heavy projectile, which make it promising in various military applications [1]-[4].

The EMCL launch process is a transient high-voltage discharging process which lasts for milliseconds. It not only concerns the varying parameters in the circuit, but also involves the rapid change in dynamics, and each other couples together tight through the magnetic field [5]-[7]. So it is very difficult to do experiment directly with the full scale EMCL prototype. The relationship of the individual quantity can only be obtained through the prototype experiment. It is hard to understand all the essential attributes of the phenomenon and unable to investigate the similar phenomenon beyond the prototype experiment conditions [8]. An alternative approach is to do research in the EMCL model experiment. The experiment results obtained by the sub-scale model can be extrapolated to the full scale prototype greatly reducing the workload and experiment expense. It is necessary to make clear the similarity relations of the physical quantities between the model and the prototype in the first instance.

The similarity theory is mainly applied in railgun in electromagnetic launching technology areas [9]-[12]. Few application is found in EMCL research. Sandia National Laboratory did some research on the working characteristics of the coilgun prototype by studying the sub-scale model in America [13]-[15]. Wuhan University in China investigated the

scaling relationship of the capacitor-driven coilgun and found that the scaling relations of the physical quantities are usually expressed as the length scaling factor and the voltage scaling factor [16]. It is hypothesized that the voltage scaling factor has similarity relation with the length scaling factor. So in this study the scaling relations of physical quantities between model and prototype in EMCL are deduced based on mathematical model of the EMCL and similarity theory. A new type of EMCL is designed and its scaling model is analysed by simulation based on the similarity relation.

II. SIMILARITY MODEL OF THE EMCL

A. New-Type EMCL Structure As illustrated in Fig. 1, the new type EMCL mainly consists of

pancake coil, drive coils, integrated launch packages (ILP) including projectile and armature, capacitor banks, switches, synchronous control circuits etc. Because of magnetic field coupling more closely between the armature and the pancake coil, the pancake coil has higher efficiency with the same exciting current. The new type EMCL reduces the coil series and the complexity of the synchronic control.

13

2

45

45

45

1

3

2

Fig. 1. The new-type EMCL model. 1- pancake coil, 2- drive coils, 3- armature, 4- capacitor banks, 5- solid dielectric switches

B. Equivalent Circuit of the EMCL Because drive coils and pancake coil have the same working

principle, they are collectively known as the launching coils in the paper. The equivalent circuit is shown in Fig. 2 when the switch of the mth launching coil is turned on and others turned off in multistage EMCL. (m=1, 2, ···, n)

978-1-4673-0305-7/12/$31.00 ©2012.IEEE

KR0 L0Rd1

Ld1

Rp

Lp

id1

ipm

Mdmpm

v

U10 D

KR0 L0Rdm

LdmidmUm0 D

KR0 L0Rdn

LdnidnUn0 D

Fig. 2. Equivalent circuit of the EMCL.

Um0 is the initial voltage of the mth capacitor banks; Cm is the capacitance of the mth capacitor banks; Um is the voltage of the mth capacitor bank at time t; v is the velocity of the projectile; R0 is the loop resistance including the resistance of the conducting wire, the capacitor banks, the switch, etc; L0 is the loop inductance including the inductance of the conducting wire, the capacitor banks, the switch, etc; Rdm and Ldm are the resistance and inductance of the mth launching coil respectively; Rp and Lp are the resistance and inductance of the armature respectively; idm and ipm are the exciting current of the mth launching coil and the eddy current induced in the armature respectively; Mdmpm is the mutual inductance between the mth launching coil and armature; D is the crowbar.

C. EMCL Prototype The physical quantity in EMCL prototype is expressed as x.

For any given time t, equations of the equivalent circuit are described as follows:

dm 0 dm dmpm pm dm 0 dm

dmpm pm m

( )d ( ) d d ( ) d ( ) ( )

d d ( )

L L i t t M i t t R R i t

v M z i t U

+ − + +

− ⋅ ⋅ = (1)

p pm p pm dmpm dm

dmpm dm

d ( ) d ( ) d ( ) d

d d ( ) 0

L i t t R i t M i t t

v M z i t

+ − ⋅

− ⋅ ⋅ = (2)

( )1m m0 m dm0

dt

U U C i t t−= − ⋅ ∫ (3)

The EM force acting on the armature along the barrel is given as follows:

( ) ( )dmpm dm pmd dzF M z i t i t= ⋅ ⋅ (4) The acceleration of the projectile is presented as follows:

( ) ( )z p dmpm p dm pm( ) d d 1a t F m M z m i t i t= = ⋅ ⋅ ⋅ (5) The velocity of the projectile is given as follows:

0( ) ( )d

tv t a t t= ∫ (6)

The displacement of the projectile is presented as follows:

( )0 0 0

( ) ( )d d dt t t

s t v t t a t t t= =∫ ∫ ∫ (7)

The resistance is expressed as follows: 0 aR l Aτ= (8)

The mutual inductance is expressed as follows: ( ) ∫ ∫ •⋅=

2 1

/dd4/ 210l l

rllM πμ (9)

The efficiency of the EMCL system is calculated as follows: 2 2

p m m 0m 1

1 2 1 2n

m v C Uη=

⎛ ⎞= ⋅ ⋅⎜ ⎟⎝ ⎠

∑ (10)

The electromagnetic field equations of the EMCL system can be written as follows: ∇× H = J (11)

t∇× −∂ ∂E = B (12) ∇ • B = 0 (13)

μB = H (14) σJ = E (15)

The stress field equations of the EMCL system can be written as follows:

ρ∇ • + ×S J B = a (16) Where mp is the projectile mass; τ is electrical resistivity; Aa is

the cross section area of the conducting wire; l0 is the length of the conducting wire; l1 and l2 are the equivalent lengths of the launching coil and armature respectively; r is the position vector; H is the magnetic field intensity; B is the magnetic flux density; E is the electric field intensity; J is the exciting current density; µ0 and µ are the permeability of vacuum and the permeability respectively; σ is the conductivity; S is the stress tensor; a is the acceleration of the projectile; ρ is the density.

D. EMCL Model The physical quantity in EMCL model is expressed as x′ . For

any given time t ′ , equations of the equivalent circuit are described as follows:

dm 0 dm dmpm pm

dm 0 dm dmpm pm m

( )d ( ) d d ( ) d

( ) ( ) d d ( )

L L i t t M i t tR R i t v M z i t U

′ ′ ′ ′ ′ ′ ′ ′ ′+ − ⋅′ ′ ′ ′ ′ ′ ′ ′ ′+ + − ⋅ ⋅ =

(17)

p pm p pm dmpm dm

dmpm dm

d ( ) d ( ) d ( ) d

d d ( ) 0

L i t t R i t M i t tv M z i t

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′⋅ + − ⋅′ ′ ′ ′ ′− ⋅ ⋅ =

(18)

( )1m m0 m dm0

dt

U U C i t t′−′ ′ ′ ′ ′ ′= − ⋅ ∫ (19)

( ) ( )dmpm dm pmd dzF M z i t i t′ ′ ′ ′ ′ ′ ′= ⋅ ⋅ (20)

( ) ( )p dmpm p dm pm( ) d d 1za t F m M z m i t i t′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′= = ⋅ ⋅ ⋅ (21)

0( ) ( )d

tv t a t t

′′ ′ ′ ′ ′= ∫ (22)

( )0 0 0

( ) ( )d d dt t t

s t v t t a t t t′ ′ ′

′ ′ ′ ′ ′ ′ ′ ′ ′= =∫ ∫ ∫ (23)

aR l Aτ′ ′ ′ ′= (24)

( ) ∫ ∫′ ′

′′•′⋅⋅′=′2 1

210 dd4l l

rllM πμ (25)

2 2p m m 0

m 1

1 2 1 2n

m v C Uη=

⎛ ⎞′ ′ ′ ′ ′= ⋅ ⋅ ⋅⎜ ⎟⎝ ⎠

∑ (26)

′ ′ ′∇ × H = J (27) t′ ′ ′ ′∇ × −∂ ∂E = B (28)

′ ′∇ • 0B = (29) μ′ ′ ′B = H (30) σ′ ′ ′J = E (31)

ρ′ ′ ′ ′ ′ ′∇ • + ×S J B = a (32)

E. Similarity Relations between Prototype and Model The scaling process is simply a coordinate transformation. If

the physical quantities in EMCL prototype and EMCL model are expressed as x and x′, then the transformation may be expressed by the following relation:

xc x x′= (33) Substituting the scaling factors of the physical quantities into

(1)-(16) respectively according to the law of similitude [17], the relations of the scaling factors can be obtained as:

dm dm 0 dm dm dm 0 dm

dmpm pm dmpm pm m

L i t L i t R i R i

M i t v M z i U

c c c c c c c c c c

c c c c c c c c

= = =

= = ⋅ ⋅ = (34)

p pm p pm dmpm dm dmpm dmL i t R i M i t v M z ic c c c c c c c c c c c= = = ⋅ ⋅ (35)

m m0 dm mU U i t Cc c c c c= = \ (36)

dmpm dm pmzF M i i zc c c c c= (37)

p dm pm dmpm p1

za F m i i M z mc c c c c c c c= = ⋅ (38)

v a tc c c= (39) 2

s v t a tc c c c c= = (40)

0 aR l Ac c c cρ= (41)

0 1 2M l l rc c c c cμ= (42)

( )p m m 0

2 2m C Uc c c c cη = v (43)

c c c∇ H J= (44)

tc c c c∇ E B= (45) c c cμB H= (46) c c cσJ E= (47) c c c c c cρ∇ +S J B a= (48)

If the length scaling factor cl is chosen,

1 2 0z s r l l l lc c c c c c c= = = = = = is got. If the materials in the

prototype and model are the same, 0

1c c c cμ μ σ ρ= = = = is set

as the preconditions. So p

3m V lc c c cρ= = is obtained. The

dimension of the coil turns N is 1, so 1Nc = is got. Because of

x y zx y z∇ = ∂ ∂ + ∂ ∂ + ∂ ∂e e e , the ∇ scaling factor is 1 1 1 1

lc = c c c c− − − −∇ = = =x y z . Substituting the scaling factors

above into (34)-(48) respectively, the relations of the scaling factors can be deduced as:

0 dm pm1

zi i F Nc c c c c c c cμ μ σ ρ= = = = = = = = (49)

1 2 0 dm p 0 dmpmz r l l s l L L L M lc c c c c c c c c c c= = = = = = = = = = (50)

dm p 0 m m0

1R R R v U U lc c c c c c c = c c c −

∇= = = = = = = =B H (51) 2

lc c c c −= = =J E S (52) 3

a lc c −= (53) 2

at A lc c c= = (54)

m

3pm C lc c c= = (55)

Seen from (49)-(55), the scaling factors of the physical quantities in EMCL are not optional, but interrelated. As long as the length scaling factor cl is given, the other scaling factors of the physical quantities are all expressed as the function of the length scaling factor cl.

III. NUMERICAL VALIDATION

A. Model Quantities of the EMCL In accordance with the scaling factors relation of the physical

quantities, model parameters in different scaling factors are computed and shown in Table .

B. Simulation Results The value of the length scaling factor cl is 1 (prototype), 2, 5,

and 10, respectively. According to the Table Ⅰ, loop current, EM force acting on the armature, velocity and displacement of the projectile are obtained through simulation as shown in Fig. 3-6. The physical quantity values are drawn comparisons in Table Ⅱ.Special Note: The horizontal axis shows time t is logt.

TABLE PARAMETERS OF THE EMCL MODEL

Component Parameter Unit Value

armature

outer diameters mm 600/cl inner diameters mm 560/cl

bottom thickness mm 30/cl axial height mm 100/cl

projectile mass kg 600/cl3

pancake coil

outer diameters mm 600/cl inner diameters mm 40/cl

axial height mm 50/cl turns 50/1

discharging time ms 0/cl2

drive coil

outer diameters mm 820/cl inner diameters mm 620/cl

axial height mm 100/cl turns 50/1

stage distance mm 100/cl discharging time of

the second stage ms 8/cl2

discharging time of the third stage ms 15/cl

2

circuit

initial voltage kV 10*cl capacitance mF 10/cl

3 resistance mΩ 10*cl inductance uH 2/cl

Fig. 3. The loop current vs. time.

-8 -6 -4 -2 0 2 40

1

2

3

4

5

6

7

8

9

t(ms)

F(M

N)

cl=1cl=2cl=5cl=10

Fig. 4. EM force acting on the armature vs. time.

-8 -6 -4 -2 0 2 40

50

100

150

200

250

300

350

400

t(ms)

v(m

/s)

cl=1cl=2cl=5cl=10

Fig. 5. Velocity of the projectile vs. time.

-8 -6 -4 -2 0 2 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t(ms)

s(m

)

cl=1cl=2cl=5cl=10

Fig. 6. Displacement of the projectile vs. time.

TABLE Ⅱ COMPARISONS WITH THE PHYSICAL QUANTITY VALUES

Physical Quantity Unit cl=1 cl=2 cl=5 cl=10 Scaling

Factor t ms 25 6.25 1 0.25 ct=cl

2 v1max m/s 26.37 52.04 130.57 259.02 cv=cl

-1 v2max m/s 30.56 60.71 153.01 307.40 cv=cl

-1 v3max m/s 36.16 72.57 181.43 358.68 cv=cl

-1 i1max kA 54.67 54.01 54.75 54.21 ci=1 ti1max ms 1.35 0.34 0.056 0.014 ct=cl

2 i2max kA 19.19 19.05 19.13 19.17 ci=1 ti2max ms 14.8 3.72 0.594 0.147 ct=cl

2 i3max kA 19.30 19.07 19.18 20.61 ci=1 ti3max ms 22 5.48 0.878 0.215 ct=cl

2 F1max MN 8.19 8.17 8.16 8.11 cF=1 tF1max ms 1.25 0.32 0.052 0.013 ct=cl

2 F2max MN 0.525 0.54 0.52 0.55 cF=1 tF2max ms 11.2 2.82 0.45 0.11 ct=cl

2 F3max MN 0.62 0.65 0.63 0.61 cF=1 tF3max ms 17.55 4.38 0.70 0.018 ct=cl

2 st mm 701.44 348.99 140.48 69.45 cs=cl η % 26.15 26.33 26.33 25.73 cη=1

Where t is the total discharging time; v1max, v2max and v3max are

the maximum velocity of the projectile in the first, second and third stage launching coil respectively; ti1max, ti2max, and ti3max are the time when the current reaches at a maximum in the first, second and third stage launching coil respectively; i1max, i2max, and i3max are the maximum current in the first, second and third launching coil respectively; tF1max, tF2max, and tF3max are the time when the force acting on the armature reaches at a maximum in the first, second and third launching coil respectively; F1max, F2max, and F3max are the maximum force in the first, second and third launching coil respectively; st is the total displacement of the projectile at time t; η is the efficiency of the EMCL system.

C. Finite Element Simulation and Results The pancake coil which has 10 turns is analysed with the finite

element method (FEM). The material properties are given in Table Ⅲ [18]. The scaled current pulse i1 applied to the model of pancake coil varying with time is shown in Fig. 3. and Table . The distributions of the magnetic flux density, Von Mises stress and deformation in the pancake coil are computed and shown in Fig. 7-9 when the excitation load reaches at a maximum. The distributions of the eddy current density, Von Mises stress and deformation in the armature are computed and shown in Fig. 10-12 when the excitation load reaches at a maximum. The physical quantity values are drawn a comparison in Table Ⅳ. They are exact images.

TABLE Ⅲ MATERIAL PROPERTIES

Material Properties Pancake coil Insulator Armature

resistivity( m⋅Ω ) 1.7e-8 - 2.8e-8

relative permeability 1.0 1.0 1.0

Young’s modulus(Gpa) 119 5.5 71

Poisson’s ratio 0.32 0.3 0.33

cl=1 cl=0.5

cl=0.2 cl=0.1

Fig. 7. The distributions of the magnetic flux density vs. cl.

cl=1 cl=0.5

cl=0.2 cl=0.1

Fig. 8. The Von Mises stress vs. cl.

cl=1 cl=0.5

cl=0.2 cl=0.1

Fig. 9. The deformation vs. cl.

cl=1 cl=0.5

cl=0.2 cl=0.1

Fig. 10. The eddy current density vs. cl.

cl=1 cl=0.5

cl=0.2 cl=0.1

Fig. 11. The Von Mises stress vs. cl.

cl=1 cl=0.5

cl=0.2 cl=0.1

Fig. 12. The deformation vs. cl.

TABLE Ⅳ COMPARISONS WITH THE PHYSICAL QUANTITY VALUES

Physical quantity Unit cl=1 cl=0.5 cl=0.2 cl=0.1 Scaling

Factor t ms 1.35 5.4 33.75 135 ct=cl

2 Bc T 7.12 3.55 1.43 0.71 cB=cl

-1 σc MPa 76 19.5 3.2 0.73 cS=cl

-2 sc mm 0.0493 0.0247 0.0099 0.0049 cs=cl Ja MA/m2 968 241 38.7 9.69 cJ=cl

-2 σa MPa 44.9 11.3 1.73 0.431 cS=cl

-2 sa mm 0.0467 0.0234 0.0094 0.0047 cs=cl

Where t is the total discharging time; Bc is the maximum

magnetic flux density of the pancake coil; σc and σa are the maximum Von Mises stress of the pancake coil and armature respectively; sc and sa are the maximum deformation of the pancake coil and armature respectively; Ja is the maximum eddy current induced in the armature.

Seen from Fig. 3-12, Table and Table Ⅳ, the scaling factors of the physical quantities in EMCL can be expressed as the function of the length scaling factor cl. The scaling method is feasible in the EMCL design.

IV. CONCLUSIONS In this study, scaling factor relations of physical quantities

between model and prototype are deduced based on its mathematical model and similarity theory. The scaling factors of the physical quantities in EMCL are not optional, but interrelated. As long as the length scaling factor cl is given, the other scaling factors of the physical quantities are all expressed as the function of the length scaling factor cl. Therefore, the performance in one model can be extrapolated to the other scaled models or prototypes. Research on the scaling model of EMCL with the similarity theory can facilitate studying the complex phenomena by the simple model experiment.

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