Transient Electromagnetic Fields

Produced by Pulsed Moving Conductors

by Ira Kohlberg, Alexander Zielinski, and CaMn D, Le

April 1999

19990419 069 :

Approved for public release; distribution is unlimited.
The findings in this report are not to be construed as an official

Department of the Army position unless so designated by other

authorized documents.

Citation of manufacturer’s or trade names does not constitute an

official endorsement or approval of tRe use thereof,

Destroy this report when it is no longer needed. Do not return

it to the originator.
I .

Army Research Laboratory

Aberdeen Proving Ground, MD 21005-5066

ARL-TR-1931 April 1999

Transient Electromagnetic Fields

Produced by Pulsed Moving Conductors

Ira Kohlberg

Kohlberg Associates, Inc.

Alexander E. Zielinski and Calvin D. Le

Weapons and Materials Research Directorate, ARL

Approved for public release; distribution is unlimited.
Abstract

A model was developed for determining the space and time behavior of the magnetic field

external to a system of moving pulsed finite conductivity conductors. The moving conductor is

in relative close proximity to a stationary conductor. The external magnetic field content has a

space and time periodicity that is related to the parameters of the system. A numerical example

predicts the magnitude and frequency of the magnetic field as a function of distance from the

stationary conductor. Potential applications of this study are discussed.

ii
‘ransient Electromagnetic Fields Produced by Pulsed Moving Conductors

lL1622618.I-I80

:a Kohlberg,* Alexander E. Zielinski, and Calvin D. Le

J.S. Army Research Laboratory

LTTNI AMSRL-WM-BC ARL-TR-1931

iberdeen Proving Ground, MD 21005-5066

AGENCY REPORT NUMBER

1. SUPPLEMENTARY NOTES

Kohlberg Associates, Inc., 11308 South Shore Road, Reston, VA 20190

2a. DlSTRlBUTlONlAVAllABlL~Y STATEMENT 12b. DISTRIBUTION CODE

ipproved for public release; distribution is unlimited.

3. ABSTRACT(hlaximum 200 words)

A model was developed for determining the space and time behavior of the magnetic field external to a system o:

noving pulsed finite conductivity conductors. The moving conductor is in relative close proximity to a stationaq

:onductor. The external magnetic field content has a space and time periodicity that is related to the parameters of the

,ystem. A numerical example predicts the magnitude and frequency of the magnetic field as a function of distance fron

he stationary conductor. Potential applications of this study are discussed.

14. SUBJECT TERMS 15. NUMBER OF PAGES

58

nagnetic field, environment 16. PRICE CODE

17. SECURlTY CLASSlFlCATlON 18. SECURITY CLASSIRCATION 19. SECURlTY CLASSIRCATION 20. LIMITATION OF ABSTRACT

OF REPORT OF THIS PAGE OF ABSTRACT

UNCLASSIFIED ” UNCLASSIFIED UNCLASSIFIED UL

. . . Standard Form 298 (Rev:2-69)

Ill Prescribed by ANSI Std. 23418 298102
INTENTIONALLY LEFT BLANK

iv
Acknowledgments

The authors wish to thank Mrs. Sharon Y. Wiley for her assistance in typing this

manuscript and in developing the illustrations. Also, a sincere appreciation is extended to

Mr. John Bennett, U.S. Army Armaments Research, Development, and Engineering

Center (ARDEC) for a concise and thorough review.

V
INTENTIONALLY LEFT BLANK

vi
Table of Contents

. . . Report Documentation Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Acknowledgments........................................................................................

List of Figures .......................................................................

1. Introduction .................................................................................................

2. Theoretical Considerations.........................................................................

3. Electromagnetic Fields and Currents in a Moving Conductor...............

4. Characterization of External Magnetic Field ...........................................

5. Numerical Results .......................................................................................

6. Summary and Conclusion ..........................................................................

7. References ....................................................................................................

Distribution List ...................................................................

V

ix

1

4

16

27

44

47

49

51

vii
IN'TENTIONALLYLEFTBLANK

. . . Vlll
List of Figures

Figure m

1. Source Current Elements and Moving Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. Geometric Considerations for Determining the Environmental Magnetic

Field Produced by a Moving Conductor System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. Single Source Current Element and Its Image for an Infinite Conductive

Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. Approximate Image Theory for Conducting Ground Plane . . . . . . . . . . . . . . . . . . . . . . . . . .

5. h:(t) as a Function of Time With Mode Index, n, as Parameter . . . . . . . . . . . . . . . . .

6. Hz(t) as a Function of Time With Mode Index, II, as Parameter . . . . . . . . . . . . . . .

7. P,(.q) as a Function of Positive q With Mode Index, 12, as Parameter . . . . .

8. Magnetic Field as a Function of Dimensionless Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

3

5

13

36

36

43

46

ix
INTENTIONALLYLEFTBLANK

X
1. Introduction

.

Over the years, there has been continuing interest in the development of pulsed-power

energy sources and accelerators based on the processes of transient magnetic field

exclusion and diffusion. An essential feature of these devices involves a stationary

source current and the generation of induced currents (i.e., eddy currents) in a moving

finite conductivity conductor. The source and eddy currents produce magnetic fields

external to their geometry that have a characteristic magnitude and frequency.

Applications utilizing this process can be found in electromagnetic braking of large

electromechanical systems [ 1, 2, 31 and the generation of high electrical power

pulses [4,5]. These applications require 100’s of k4 with millisecond pulse-widths.

Additionally, these currents can produce external magnetic fields of a few Tesla.

As these electrical devices mature into components for military systems, we must

consider the electromagnetic compatibility (EMC) issues that arise from the

“environmental” electric and magnetic fields they generate. By “environmental” fields,

we mean those fields that are created in spatial locations that are not especially close to

the regions that they originate from and are concurrently located in close proximity to

electronic equipment that may be affected by them. For example, we have found that

environmental electromagnetic fields generated by capacitor-driven pulsed-power sources

could be important EMC considerations for electromagnetic railguns [6, 71. Fortunately,

standard shielding techniques, using conducting materials, are adequate for managing all

but the largest magnetic fields (i.e., < I 7) [8]. For large confined fields, active shielding

techniques can be more efficiently implemented [8].

The purpose of this initial assessment is to characterize the salient features of the

environmental magnetic field generated by a moving conductor system that is produced

by a spatially periodic time-varying source current. Figure 1 shows a close-up of the

relationship between the spatially periodic source currents and the moving conductor.
?’

I

-L ,:

I

I

Top of moving conductor (y - A)

.

Source current

l!iifa Into the paper

Out of the paper

Figure 1. Source Current Elements and Moving Conductor.

Figure 2 shows the geometry used for determining the environmental magnetic field.

The object of ‘this study is to determine the time behavior of the magnetic field at

point P(x,,y,,z,) due to the source currents. Jn Figure 2, the system is assumed to be

periodic and infinite in the k x -directions. We have cast the problem in Cartesian

coordinates for mathematical simplicity, but, with the proper mathematical

transformation, it can be made to correspond to cylindrical geometry.

In this report, no attempt is made to relate the test results to a specific application. A

significant amount of detailed information would be required to accomplish this, and such

an effort is beyond the scope of the present study.

The results of our model quantify the damped harmonic electromagnetic fields

generated by a moving conductor in terms of the periodic spatial separation between the

current sources, the conductivity and velocity of the moving conductor, and the location

of the observation point relative to the moving conductor. The principal contributions to

the far-field magnetic signature, however, are shown to be due to the source current

distribution and its time-dependent image created during a transient pulse.

2
A

Moving cond&, p, D

;. a ‘. +

: ._

‘_ X

-

w

w

-

--IFigure 2.

. , . Field point, P ‘0 1 (x,,, y,,, ~2

Front View

A,

A ,, ._ . . : “’

‘_, h;. .’

:_‘ : s’

:

‘,

Side View

Geometric Considerations for Determining the Environmental Magnetic

Field Produced by a Moving Conductor System.

The mathematical details leading to the determination of the current density in the

moving conductor and the subsequent prediction of the environmental magnetic field are

tedious to follow. In order to facilitate the process of understanding the basic physical

ideas, analysis, and interpretation of the results, a brief overview of the basic physical

processes is presented in section 2. The theoretical prediction of the current density and

associated magnetic fields in the moving conductor is rendered in section 3. Using the

results of section 3, the environmental magnetic field is computed in section 4.

Numerical results are rendered in section 5. Summary and conclusions are presented in

section 6.

3
2. Theoretical Considerations

Much insight into the structure of the electromagnetic fields described in Figures 1

and 2 can be gleaned by first considering the solution for a single-source current element.

For convenience, we select the center source element of Figures 1 and 2. Elucidation of

the essential physical phenomena is further facilitated by considering the two-dimensional

case in which source current elements are infinite in the z-direction and where the

thickness, A , is suitably large. The implications of this latter assumption are defined in

the course of the discussion.

For illustration, we begin with the case in which the conductor’s velocity is zero. We

want to determine the response of this nonmoving conductor to a sudden increase in the

source current, I(t) : t 2 0, which flows through each source element. (The origin of I(t)

is not essential for understanding the issues related to the electromagnetic fields.

Nevertheless, it is worthwhile to mention that, in the case of an energy-producing device,

I(t) is determined from external considerations that involve: generation of an

electromotive force in the stationary conductor, an electrical load, and the current induced

in the moving conductor by I(t) itself. For example, in the case of a compensated pulsed

alternator [5], the effect of the moving conductor is to produce a time-dependent

inductance. Detailed discussions on the types of compensation and the methods

physical realization of compensation in pulsed machinery can be found

literature [5].)

The problem of the stationary conductor further divides into two categories:

mutual

for the

in the

(1) the

case where the conductivity of the moving conductor is infinite and (2) the case where the

conductivity is finite. A close-up of the situation for a single conductor with CT = 00 is

shown in Figure 3.

4
Current

Into the paper (image)

Out of the paper (source)

-

I I

1, -

Figure 3. Single Source Current Element and Its Image for an Infinite Conductive

Surface.

The basic physical law that determines the current flow in the case where CT = 00 is

concerned not with the magnetic field, but with the electric field. The surface impedance

for a medium that has infinite conductivity is identically zero. This requires that the total

component of the electric field on the surface, which is parallel to the surface, EN, must

also be identically zero. A reflected electromagnetic wave must therefore have its

component of the surface electric field that is parallel to the surface exactly cancel the

corresponding component of the incident electric field [9]. The only component of the

electric field that can exist on the surface of a perfect conductor is the component that is

normal to the surface, E, .

The foregoing discussion concerning E, applies at all frequencies. It can be shown

using the theory of a horizontal dipole over a conducting half space, taken in the limit of

CT = 03, that the condition E, = 0 leads to image theory [9]. When applied to Figure 3,

image theory shows that the magnetic field at the surface of an infinite conducting body

can be computed from two sources [9]. The first source is the true physical current

source (flowing out of the paper), and the other is the image of the source current that

flows into the paper.
Image theory permits the field at all points in space to be computed by summing the

fields due to the actual physical real source and its image. The real source and its image

each produce a magnetic field determined from the Biot-Savart law [9]. The three

dimensional form of the Biot-Savart is given by

(0

where yC is the current density; I?, 7, Z are the coordinates of the volumetric distribution

of TC; and x, y, z are the coordinates at which the field is to be computed;

T=T(x-X)+j(y-jq+i(z-Z)

and

(2)

II (

r’ 3 = (x-q2 +(y-jy +(z-T)2)3’2, (3)

where r, 3, and k’ are unit vectors in the x-, y-, and z-directions, respectively. For

currents that flow only in the z-direction, we have

-__ 2 = j,(&Y,Z)Z (4)

and

/cxr’= j(x-q-i(y-7). (5)

When applied to the two-dimensional configuration of Figure 3, the contributions

from the real source, & , and image, 8, , are given by

6
and

where

I(t) J,(t) = 7

1

(7)

(8)

is the current per unit length of the source current element. When equations (6) and (7)

are evaluated on the surface of the conductor, y = 0, we find that the y-component of the

magnetic field vanishes and the x-component, BSx, consists of equal contributions from

the real source and its image. We get

Integration over 7 gives

B,,(x) = = B,,(-x) . (10)

Using the principle of superposition, the total surface magnetic field corresponding to the

alternating array of Figure 1 is

7
By making the substitution x = x + mL , where m is any integer, in equation (1 l), we

easily see that Bsc is a periodic function of L. We derive

II’=+=3

Bs;(x+mL) = c 13sx(x-~)(-ly~,

n’=-UA

(12)

where the new dummy variable of summation is now n’ = 2m - n . Since n and n’ are

summation indices, equation (11) is the same as equation (12).

Combining the symmetry and periodic properties of BsT, we can write

n=ce 2n7rx lls; = CB,:(t)cos~,

ll=l

where

(13)

(14)

The modal components, Bs=., are determined by straightforward integration of

equation (14) using equations (10) and (11). The details of the calculation are not

important. It is important to note, however, that all Bs! are proportional to the product of

- p0 Jo (t) . Accordingly, it is convenient to use the form

(1%

8
where the Sz, are shown to be dimensionless expansion coefficients given by

I

dx. (16)

The magnetic field intensity at the surface due to all the conductors is

(17)

The general expression for the surface current density, 5, is given by the vector equation

where BS is magnetic field intensity on the surface and n’ is the normal to the surface of

the conductor. As shown in Figure 3, n’ points in the negative y-direction and

Inserting equation (19) into (18) gives

i$ = -ti,(r)ny$, co,?.

n=l

(20)

In contrast to the discontinuous elements of the array shown in Figure 1, the current

density on the surface is always continuous because the surface is continuous. For

example, in the case of a single element, which is discontinuous, the surface magnetic

field has been shown in equation (10) to be continuous. Using equation (10) in

9
equation (18) then provides a continuous surface current. Discontinuities in surface

current lead to singular point accretions of charge on the surface. This does not appear to

be physically plausible in this problem.

Equation (20) shows that 3, is a periodic function of x. By comparison, the

discontinuous current density, yR , of the periodic distribution of source elements shown

in Figures 1 and 2 can be written as jR = CR. The magnitude is

2nnx J&t) = &$(t)cosy-,

n=o

(21)

where

L=W,+l,), (22)

and the 4 ‘s are determined from the orthogonality properties of the Fouier series. Using

the discontinuous distributions

J(x,O = J,(t) :~rlxl20,

and

10
=-J,(t) :z,+z2 21x12$+&

we derive

4, =%(I-cosnn)sin%.

(23)

(24)

The image distribution for the array of source elements is the negative of equation (21).

In contrast to the image current, which is a mathematical abstraction, the surface

current of equation (20) is a real current. However, from the point of view of calculating

the conductor’s contribution to the magnetic field for y c 0 from the Biot-Savart law, we

have a choice of using either the real surface current located at y = 0 or the image current

located in the plane y = w.

In summary, we have learned the following from the infinite conductivity case that

has bearing on the general case involving a moving conductor having finite conductivity.

(1) A true physical surface current is produced on the conductor that is generated by

the source current array.

(2) The physical surface current is computed from the source current and its image.

(3) The surface current is continuous.

(4) The time behavior of the surface current and its image is proportional to the time

behavior of the source current. For example, if the source current were to remain

11
constant in time, so would the surface current and the image current. The reason

for this is that infinite conductivity precludes energy dissipation and, hence, a

current once generated will persist indefinitely (in the absence of radiation losses).

We now consider a nonmoving conductor with finite conductivity. This extension is

one that involves an enormous increase in complexity as compared with the infinite

conductivity case. As previously mentioned, the basic physical problem to be solved here

is that of a horizontal dipole over a conducting ground plane. This problem was first

attacked by the German theoretical physicist, Arnold Sommerfeld, at the beginning of the

20’ century, and, for the fast half of this century, played a crucial role in the field of radio

science. Analytic study of the famous “Sommerfeld Integrals” connected with the

solution continues to this very day. The seminal text by Banos [lo] provides a

comprehensive discussion of this fundamental and difficult problem.

The solution for the horizontal dipole over a conducting ground furnishes a complete

space-and-time description of the electric and magnetic fields both inside the conductor

and in free space. For example, Hill and Wait [ 111 solve explicitly for the fields in the

case where the line source lies directly on the ground. Our problem is simply an

extension of the basic problem to include a finite number of distributed elements.

Although this type of rigorous formulation can be attempted, it is not worthwhile to do so

unless we can demonstrate that simpler models can’t do the job.

The literature on horizontal dipoles over ground planes is extensive and has been

addressed by the leading electromagnetic theoreticians in this century. A major outcome

of these studies has been the emergence of an approximate image theory that is applicable

for conducting media [12]. Figure 4 shows how this theory is applied. If w is the

location of the image for 0 = ~0 , the total displacement, wC , for cr f 00 is given by [ 121

WC =w+d, (25)

12
Figure 4. Approximate Image Theory for Conducting Ground Plane.

where d is the additional displacement due to the conductivity. In frequency space, d is

actually a complex quantity, a consequence of fitting the rigorous theory with a

mathematical approximation [ 121. It is given by

d = 2

(jw &&* ’

(26)

which is valid for the condition, o >> o E (applicable in our case).

If we associate the magnitude of d with the physical displacement and let o = 27r / tp ,

where tp is the duration of a pulse, we obtain

(27)

Equation (27) is (within a numerical constant of order unity) the root-mean-square

displacement determined from diffusion theory. If tp is small, the displacement of the

image will likewise be small. Equation (27) thus provides the physically intuitive result

that, for short times, the location of the image is that obtained from the cr = 00 case. In

13
14

order to apply equation (27) with any degree of confidence, the thickness, A, must be

greater than IdI.

In order to use image theory, we need to know not only its displacement, but also the

magnitude of the image current. Whereas in the CT = ~0 case, the magnitude of the image

current is equal to that of the source current, this will not rigorously be true in the finite

conductivity case. The magnitude of the image current will be less than that for the

infinite conducting case because there are energy losses in the system associated with

finite conductivity. The rigorous solution of this problem requires the formulation in

terms of a frequency-dependent horizontal dipole over a conducting half space [lo].

When the velocity is not zero, the location of the image is the same as that for the

stationary conductor for CT = ~0. The reason for this is the same as before: the only way

we can assure zero electric field at the surface is for the image to be opposite to the real

source. Since the source is stationary, so is the image.

An explanation of current behavior when the conductor is moving and 0 f 00 is not

easily explained without solving the entire problem. Conceptually, this problem can be

attacked as a moving horizontal dipole over a conducting half space. To our knowledge,

this approach has not been attempted. By all reckoning, it would appear to be an

extremely difficult problem. Nevertheless, this is the problem that is addressed in this

study.

Our solution is based on solving Maxwell’s equations,

Vxa=aiz+avxE ,

VxiL--;,

CW

Wb)
(28~)

and

V&O (284

I ’

in a conductor moving with velocity, V . In our model, it is assumed that the conductor’s

thickness, A, is large enough so that the electromagnetic fields never diffuse out of the

moving conductor. Mathematically, the large enough assumption means that we can set

the magnetic field components equal to zero at y = A.

The foregoing assumption about A being large has only an indirect bearing on the

calculation of the environmental magnetic field from the currents in the system. What is

really important is the depth of penetration of the current into the moving conductor as

compared with the vertical distance of the field point at which the magnetic field is being

calculated from the conductor surface. As we show in this study, the depth of magnetic

field penetration is quite small compared with meaningful distances.

An accurate prediction of the fields is possible, provided that we can precisely specify

the magnetic field at the bottom surface for all time, t 2 0. This is exactly where the

difficulty arises; the magnetic field at y = 0 cannot precisely be defined for all time

without simultaneously solving Maxwell’s equations in the space below the moving

conductor.

It would appear that an approximate solution to this problem can be obtained using

perturbation theory. The concept is as follows: first, we solve for the electromagnetic

fields in the conductor by assuming that the magnetic field at y = 0 is given by image

theoryfor 0=~. This approximation enables us to determine the first-order correction

for the displacement of the image due to diffusion and the decrease in the magnitude of

15
the image current due to energy losses associated with fmite conductivity. Corrections to

the surface magnetic field are then made.

3. Electromagnetic Fields and Currents in a Moving Conductor

Maxwell’s equations in a moving conductor are given by equation (28). In the

geometry of Figures 1 and 2, the velocity is given by the relationship: v’ = Fv. By taking

the curl of equation (28a) and making the substitutions of equations (28b) and (28~) for

Vd and I? in equation (28a), we derive the following identical equations for B, and

By:

and

(29a)

VW

Equations (29a - b) are, however, not entirely independent. The divergence condition

of equation (28d),

(30)

creates an interdependency on the solutions of equations (29a - b). They are each secondorder in x and y, and first-order in time. If it were not for the divergence relationship,

they would be completely independent. For the boundary conditions of our problem, we

16
readily show that the solution of equation (29a) combined with equation (30) uniquely

determines BY.

Let us first consider the solution for B, . We must specify the boundary conditions at

y=Oandaty= A. The boundary condition at y = A is greatly simplified for pulse

lengths, tp, that are short (i.e., A >I d I). When these conditions apply, we are at liberty

to set both components of the magnetic field equal to zero at the top boundary:

B,(x,y = A,t) = B,(x,y = A,t) = 0. (31)

This is the boundary condition used in our study.

The foregoing boundary will be relevant only for those cases where the diffusion

time, to, from the initiation of the pulse at y = 0 to y = A is larger than tp. .The

diffusion time is approximated by the formula

to = 6~~0. (32)

Diffusion time can vary greatly, depending on the application. For example, using an

aluminum conductor ( o=2.W07 mhoslm, &,=47r~lO-~ H/m), and a nominal thickness

A= 1.5 cm gives

to= 6.2 ms. (33)

When tp is significantly shorter than to, the top boundary of Figure 1 is never

reached during its duration and, for all practical purpose, it could be located at y=a . On

the other hand, for a system in which there is a continuous stream of pulses, the condition

at the top boundary may need to be more critically examined.

17
The solution of the problem requires that we specify the time behavior for B, at y==O

for t 2 0. This quantity is .denoted in this section as

Bx” = B&y = 0,t). (34)

B: is determined from the considerations rendered in section 2. For example, we can set

(35)

which is the field due to the periodic source of Figure 1 over a perfectly conducting plane.

Use of equation (35) constitutes the fast approximation in a perturbation scheme.

Our analysis, however, ‘is carried out in general terms by keeping the spatial

periodic&y and allowing for an arbitrary time behavior in J,(t) . This is accomplished

using a Delta (impulse) function current density source. The boundary condition at y = 0

then becomes

B,D = - j@(t)E a,, COST. (36)

R=l

If B, and B,, are now interpreted as the solutions to equations (29a) and (29b) for the

impulse function, the actual fields, B:(x,y,t) and Bf(x,y,t) will be given by

and

Wa)

18
W’b)

In section 4, we apply the foregoing formulas to the case where J,(t) is a step function

that has the properties

Jo = AJ :t,a20 WW

Jo =0 :tq, (3W

where AJ is a constant.

Denoting the Laplace transform variable by “s” gives

where ix is defined as the Laplace transform of B,. The foregoing equation is next

reduced to one involving only the y-dimension by using the periodicity of the x-direction.

We have

B’,(x+L,y,s) =&xy,s).

Using equation (40), we next write ix as the real part of the harmonic series

w9

(41)

where

19
2nZ

Ynn’y-. (42)

Substituting equation (41) into equation (39) and equating terms proportional to ejynX

gives the following differential equation for x(y,s) :

(43)

The solution for 7” is

where

(4-J)

(45)

Since the magnetic field must go to zero at y=m, we require that & =0 and thus

have

jn = FnemknY . (W

Inserting equation (46) into equation (41) gives

(47)

Using the same methodology for equation (29b), we derive the following expression for

By:

2 0
Ijy = n=C+p-(S)e-hzYejynx, (48)

n=-ca

where p,(s) is related to in(s) through the divergence relationship of equation (30):

(49)

The only way that equation (49) can be satisfied is if the “(

Setting these terms equal to zero gives

)” terms are equal to zero.

(50)

From the foregoing equations, we see that the Fn ‘s determine the x- and y-components of

the electromagnetic field. The Fn ‘s are determined from the boundary condition of

equation (36) applied in Laplace transform space. Recalling that

z 6(t) = 1, (51)

where I? is the Laplace transform operator, and evaluating equation (47) at y = 0 gives

B; =-&pin cozy = gFn(s)efin’ . (52)

n=l

Using eiyn* = cos y,x + jsin ‘y,x gives

ll=+m lZ=+CO

C <(@e&n” = e + C [(E + E_,)cosy,x + j(e - F,)siny,xl . (53)

21
Inserting equation (53) into equation (52) and noting that there is no Q, term in

equation (52) gives

WO

F* -3, =o, WV

and

E=o.

From equations (54a - c), we have

w4

(55)

The time domain x-component of the magnetic field is obtained by taking the inverse

Laplace transform of equation (47):

where F,(t) is the inverse Laplace transform of Fn (s) and

Q,(z) = z-l(@“) = z-‘&(S).

Q,(z) is computed in two steps. First, we write

k, = ,&i&/~,

22

(56)

(57)

(58)
where

Then

q

R

(s)= ,-(Gx)

.

From the properties of the inverse Laplace transform, we have

where

Using the formula

then yields

z-l(e-aJ;) = Le-a2/4t =y(y,t)

2Jz

Q,(z) = e-‘n’“,-g = e-“7y(y,z)

2J,tj;

: y > 0.

(60)

(62)

(63)

(64)

Equation (56) is simplified using the result

23
Substituting equations (64) and (65) into equation (56) yields

Using the formulas

and

I-n =aL_jqY, Y2 =r;,

PO0

-2m

Y-n = -Yn=L9

reduces equation (66) to the simpler forms

and

2

B, = -poT ~,W(y,r)e-~cOS(Ynx-yV~)

n=l

6%

(67W

(67~)

(69)

24
We now compute &(x, y,t) by taking the inverse Laplace transform of equation (48).

Using equations (50), (54a - c), and (58), we initially have

followed by

where

(70)

(71)

r-,(z) = 2-l

t&

,-(JiGqXY) 1

.

n

Using equation (61), we get

WY2

= e-'"7 - e

l-47

$_

= e-“‘@(y,z),

nr

(72)

(73)

where
aqy,z) +e--P*UY z

d- 4T . (74)

Substituting equations (47), (73), and (74) into equation (71) and then integrating over r

gives

Since “/, is an odd function of 12, while Q, is and even function of n, we get

Combining positive and negative values of n, and using equation (76) gives

Ytf Qtfe fln(x-vf) + y_nQ_ne+jy_n(x-yf) = 2 jy,sZ, Sh(y,X - y,Vt) .

Inserting equation (77) into equation (75) yields the following expression for B,, :

(75)

(76)

(77)

(78)

Lastly, the current density in the moving conductor is determined from the formula

(7%
4. Characterization of External Magnetic Field

I .

The purpose of this section is to calculate the environmental magnetic field in the

region below the source conductors (see Figure l), that is, for field points satisfying the

condition yO < -w . The magnetic field at the point x0, yO,zo is determined in a

straightforward way using the Biot-Savart formula of equation (1) with x, y, z replaced

by x0, y,,.zO . The volumetric integration in equation (1) is over all current densities.

In our problem, there are two current sources that contribute to the environmental

magnetic field: the first is the source from the input current source elements and the

second is from the spatial distribution of current density within the moving conductor.

Using the formula

VxZ=k , (80)

the z-component of current density in the moving conductor is given by

The z-component is the only component of current density in our system and is a

consequence of the geometry. Using the results of section 3, which provide B, and B, as

a function of x, y, and t, we can calculate the spatial distribution of current density for the

moving conductor. When the resulting expression for j, is inserted in equation (l), the

environmental magnetic field from the moving conductor is determined as a function of

x,,,y,,,z,,, andtintheregion yO C-W.

27
In contrast to the exact computation of the environmental field from the spatial

distribution of current density within the moving conductor, the contribution from the

source current is relatively easy. The y-dimension of the source current is assumed to be

very thin compared with IyO - ~1. For the moving conductor, the vertical integrated

current (i.e., image) is

Placing J * at some equivalent height above the ground plane y = 0 is analogous to placing

an image current a distance w, = w + d above a finite conducting ground plane.

From the infinite conductivity case, we know that the current in the conductor will act

to compensate the magnetic field generated by the source current. For this CT = 00 case,

the integrated vertical current is a continuous planar distribution, 180” out of phase with

the source current. This image is located a distance w above the y = 0 plane. The

resultant environmental field will be the smallest in this case because it is composed of

two parts that nearly cancel each other when lyol < -w .

When CT # 00, the image current cannot be placed any closer to y = 0 than w. This is

true irrespective of the velocity. In the other extreme, we can place the image current so

far above the y = 0 plane (this is only a mathematical abstraction) that the environmental

magnetic field will be due to only the source current elements.

In summary, the upper bound on the environmental magnetic field will be determined

by completely neglecting the current in the moving conductor. The lower bound will be

computed by locating the image current a distance w above the y = 0 plane. We choose

to render the results in this format, not because we cannot calculate the distant magnetic

fields exactly [ &can be calculated exactly from equation (81) for use in the Biot-Savart

28
law], but because this viewpoint provides enhanced insight into the salient physical

processes.

Equation (82) can be formally integrated to yield

J* = -$G(x,t) -B,(x,y = qt) + B&y = O,t)], (83)

0

where

G&t) = j aqj;x;y’r) dy .

0

(84)

Taking the partial of equation (78) with respect to x and then integrating over y gives

n-w_&

G(x,t) =f ze Pou y2Q, cos(y,x - ypt) . n

n-1

(85)

The exponential terms in equation (85) are a result of diffusion in the x-direction. The

modal diffusion time constants are given by

t&t _ w _ POOL2

r,’ 4n2fr2 ’

636)

using /lo -- 47r x I 0m7 H/m, o = 2.2 x 1 O7 mhoslm, and a nominal dimension of L = 0.5 m

gives

tdl = 0.18s (87)

29
for the first mode.

they will generally

millisecond range.

completed.

Even though higher-order modes will have shorter diffusion times,

be much longer than the pulse lengths of interest-typically in the

We return to this time-constant issue after our general analysis is

Returning to equation (83), we note that since B,(x,y = a,t) = 0, we can use

equations (34) and (35) to write

J* = G&t) + B&Y = od G(xJ) *=m

PO

---s(t)& COSY,X *

PO - l-4 n=l

(88)

Because all the magnetic field components determined up to this point were based on

the impulse function, so is J*. Applying the integral relationship of equation (37) as

applied to J' , the step function response for the total current per unit length is

When equation (88) is inserted into equation (89), we get two contributions, which

are distinctly associated with the image contribution described in section 2 and another

part that is identified with diffusion and motion. We write

J*Step =J;IM-Jj, (90)

where the image contribution is given by

J; = A,jS(z)d~~~~ cosy,x = A, y&2, cosy,x,

0 n=l n=l

(91)

and the diffusion-motion contribution is given by

30
J*D M (92)

Equation (92) is evaluated using equation (85). Before presenting the general result, it

is instructive to evaluate JiM for the special case of a stationary conductor, v = 0. These

results confirm the assertions rendered in section 2. When v = 0, we have

1 n=+o3 22

G(x, t) = -ce p~“r,“Q, cOSy,x ,

CJ n=l

and

J*D M =~jG(x,r)dr= AJzC2,(l-e -$cosjfnx.

P0 0 ?I=1

In carrying out the integration in equation (94), we have used the formula

(93)

Equations (91) and (94) are rich in physical content and can be traced back to the

discussion of section 2. If we let CT = 00, then tdn = ~0, all the “( )” terms of

equation (94) vanish, and we are left with only the image contribution,

J* =-J;, step (96)

By comparing equation (94) with equation (91) on a term-by-term basis and noting

that

3’1
(l-e&l, (97)

we readily show that, on the average, over the periodic spatial interval (0, L}, the

magnitude of Ji,,, is less than the magnitude J; . The proof is straightforward using the

orthogonality of the cosine terms. We have

and

(98)

(99)

For all real physical systems, IJ~~av, must be bounded and, hence,

A2 n=m ( 1

l/2

t=,

c

sz; <m. (100)

Every term in the series of equations (98) and (99) is positive. Since each term in

equation (99) is greater than the corresponding term in equation (98), the proof that

kl’l,> IJLMImve is complete. In the regime where t cc t&, we have the approximation

1 _ pd” =tlt, <<l. (101)

In this case, image theory provides a very good description of current flow.

32
Lastly, we notice that equations (91) and (94) can be inserted in equation (90) to give

the result for a stationary conductor based on a first-order perturbation to the perfect

.

conductor solution at y = 0

n=m

J*SW = -A, Cf2”e-t cosy,x. (102)

II=1

Equation (102) shows that the current in the conductor will eventually vanish for a step

function excitation and is due to conductive energy loss.

We now consider the general case involving a moving conductor, which requires use

of equation (85) in equation (92). This calculation is facilitated by first writing

cos(y,x - y,vt) = cos(y,x) cos(y,vt) + WY,X) sin(lw) *

Inserting equation (103) into equation (85) gives

and

G(a) = G,(x,t) + G2(x,t), (104)

where

*+m~

G,(x,t) =; xe RyQn cos(y,x) cos(y,vt>

n-1

lr+m -_&

G&t) =; se “r,‘Qn sin(y,x) sin(y,vt) .

II-1

Substituting equations (104) - (106) into equation (92) and integrating yields

33

(103)

(105)

(106)
where

and

1 Y,2J

Ym =

((r,’ / &CT>” + (y,d2) ynv - e

( (

-Tz A

Y” sin(y,vt) + Y”V cos(y,vt) * (10%

Jw 11

The foregoing expressions show that the conductor current contains harmonics with

angular frequencies:

2n7cv

w, =y,v=-L * uw

Using values of L = 0.5 m, and v = 200 mls gives

W, = 2,500 radls uw

for the first harmonic.

The damping time constants given by equation (86) show that the higher harmonics

dampen out much more rapidly than the lower numbered ones. This damping produces

frequency broadening, which must be taken into account when analyzing the response of

electronic equipment that is located near the system of conductors shown in Figure 1.

34
In order to better understand the essential features of environmental magnetic fields

generated by the moving conductor current system of Figure 1, it is desirable to compare

the magnitude of the modal terms in equation (107) with the corresponding modal terms

of J; as given by equation (91). We shall use the same procedure for making this

comparison as we did in the development of equations (98) and (99).

Using the orthogonality of both cosine and sine function in computing IJLI, from

equation (107), we obtain

where

Y2 Y2 Hi =+++.

tdn t&l

(112)

(113)

With the help of basic trigonometry, algebra, and equations (108) and (log), we

readily prove that

Hf = ”

n2 + (t&J) 2 h, 0) 3

where

21 t

h,(t) = l+ e-‘” - 2e-; cosqt .

(114)

(115)

35
Figures 5 and 6 show h,(t) and Hi(t) , respectively, as a function of time with mode

number, n, as a parameter for p0 = 4~ x IO7 Hhn, o =2.2x IO7 mhoslm, L = 0.5 m, and

v = 200 mls.

1.5

hn

1

0.2 0.4

Time (s)

0.6 0.8

Figure 5. h, (t) as a Function of Time With Mode Index, n, as Parameter.

0.00012 -

0.00010 - !:

!i

0.00008 - /:”

I!&,* 0.00006 - i: ‘:

___________________---___-_--- _____ ________ _ ____ _ ___-_____ _--

0 0.2 0.4 0.6 0.8

Tb (s>

Figure 6. Ha(t) as a Function of Time With Mode Index, n, as Parameter.

36
By comparing equation (112) with equation (99) on a term-by-term basis, we note that

their ratios are HE. Equation (115) shows that the maximum value of h,,(t) is four.

Thus, for this exploratory study, with the mode index, n, less than 10, Hz can be

approximated by its upper bound as

Hf = &12

n2 + (tdlWl)2 ’

(116)

We also have

tJ0, = 0.18 (2,500) = 450. (117)

The higher order terms of IJlfM loye become comparable to those of IJII, when Hf

approaches unity. Equations (116) and (117) show that this will occur when the mode

index becomes greater than

= 260.

Considering the first few modes, I&I,,, will be significantly less than IJ;I,. The

motion of the conductor does not appreciatively affect the environmental field.

For the purposes of calculating and bounding the environmental magnetic field in the

region y0 < -IV we need to consider the real current density source located in the plane

y = --w and its image current density source located in the plane y = +w (see Figures 1

and 2). As previously mentioned, the maximum environmental magnetic field will occur

when the image contribution is absent and the smallest will occur when the image is

37
‘

where the integration over x extends from - ~0 to + ~0, the integration over z extends

from -h to +h:

included. These two limits bound the problem, irrespective of how well we calculate the

details of the current in the moving conductor.

The starting point for the calculation of the magnetic field at position

(119)

is the Biot-Savart law, which is given by equation (1). In order to be consistent with the

notation of section 3, we let x, y, z be the coordinates of the volumetric distribution of

current density; the reader may recall that, in section 2, X, y,Z were used for the

volumetric distribution of current density. When we consider only the real source and its

image, the magnetic field at < is given by

where & and gI are the contributions from the real and image sources, respectively.

They are given by

and

38
and

.

T,=qx,-x)+j(y,+w)+k(z,-z),

kxr;,=~(x,-cc)-i(y,+w),

~~~I)=((x~-x,‘+(y~+w)*+(z~-z)*)31*,

ij =i(x,-x)+j(y,-w)+i(z,-z),

ix6 =j(x,-x)-i(y,--w),

-3 II (( rr = X0 -x)'+(yO-~~*+&- z)*Y'*.

(123a)

(123b)

(123~)

(123d)

(123e)

(123f)

In equations (123a - f), we place no restriction on the location of the field point, although

we apply the results to those locations where y,, < -w .

The negative sign in equation (122) and the appearance of JR in the integral accounts

for the fact that the image current is the negative of the real current. Equations (123a - f)

account for the fact (see Figure 2) that the source current density is located at y = -w ,

while the image current density is located at y = +w. JR is determined from

equations (21) and (24), with Jo given by the step function of equation (38). We have

JR(x) = ~Qos~

n=l

(124)

and

39
4, = %(l- cosnn)sin%. (125)

The calculation of the analytical forms for & and & is straightforward. For brevity,

we calculate & and then simply determine 8, by substituting y, - w in place of

y, + w and taking the negative of the result. Equation (121) is first integrated over z to

yield

= -?(y, + w)W + 3<X0 - X)W,

where

W=i” dz

-h(b* + (Z, - Z)2)3'2’

and

(126)

(127)

(128)

The integration in equation (126) gives

,2 z, +h z/h

b2 ((zo + h)2 + b2)1'2 - ((z, - h)2 + b2y2 1

-

(129)

The foregoing expression simplifies when we take our observation point to be located

at the midpoint, z, = 0. There is essentialli no loss of generality in this case, and W

becomes

40
(130)

Since one of the principle aims of this study is to examine the relationship between

the distance of the field point from the conductors as compared to the spatial periodicity

of the current in the moving conductor, we further simplify W by setting x0 = 0.

Equation (130) then reduces to

2h 1

w =

x2 + (Y, + WI2 (h’ + x2 + (y, + w,,l” ’

(131)

Equation (13 1) is used in subsequent calculations.

The procedure from this point is to substitute equations (124), (126), and (13 1) into

equation (121) and then integrate over x between the limits of - ~0 to + 00 . Formally

integrating over x initially yields

(132)

Since JR and W are symmetric functions of X, and the “x” in the second integral is an

antisymmetric function of X, the second integral vanishes. Inserting equations (124) and

(131) into equation (132) gives

&=-~ n=e¶

t(y, + w)x@Jyo + w,h) 9

tl=l 1

(133)

where

41
0, = j&‘““x 2h 1

L X2+(Yo+wj2 (h2+x2+(yO+w)‘)l/2 ah-

-ce

(134)

Simplification results when we cast the foregoing equations in dimensionless form by

introducing the variables

P=$

Yo + w v= L 9

(135)

(134)

and selecting a specific value for h. For convenience a value of h = L is used.

Using equations (125), (135), and (136) we derive the following expression for BR ,

recalling from equation (133) that it is the x-component (hence we drop the vector

notation):

BR =

where

=-&(l- ml,

% cosm) sin-L

and

112 dP* (139)

42

(137)

(138)
Figure 7 shows P, as a function of positive q , with mode index, n, as parameter. As

expected, P, decreases with increasing distance from the moving conductor and with

increasing mode index. We also have

w17) =--p,(q),

which is relevant in our calculations since y, is negative.

W)

-n=l

- - n=2

_ _ _ _ .n=3

0 0.2 0.4 0.6 0.8

rl

Figure 7. P,(T) as a Function of Positive q With Mode Index as Parameter.

The total magnetic field is computed from the formalism of equations (120) to (123).

In terms of the field point used in the foregoing calculations x0 = 0, z, = 0, we have

the second term being attributed to the image.

I ’

43

(141)
5. Numerical Results

The purpose of this section is to obtain numerical estimates of the magnetic flux

density as a function of vertical distance from the source conductor system for a particular

set of parameters. The following values are used:

I, = 17.5 mm,

1, = 7.5 mm,

L=2(Z,+Z,)= OSm,

w = 2.5 mm,

h = L,

and

I, = 600,000 A.

There is a wide range of magnetic field variations in this system. The highest fields

occur very close to the source elements themselves. In this region, which is not

anticipated to be close to any external electronics systems in practical applications, the

magnetic field intensity is of the order of

H =O=I A, = 3.4 kAlmm

4

(142)

and the magnetic flux density is

44
B = lu,H = 4.2 T. (143)

As we move away from the individual conductors, the influence of the periodic array

of the source current is seen and the theory provided in the previous section must be used.

The largest fields in this case exist when the image effect is negligible. This occurs when

the image distance becomes large. As previously discussed, the case in which the image

currents can be neglected is a mathematical abstraction and provides a limiting case of

maximum environmental magnetic field. This field is given by

B,, = BR = -$AJnTa,P,(q).

n=l

WJ)

Using the numbers provided at the beginning of this section and noting from

equation (138) that

an = 0 : n = even number, (145)

we obtain al = I.15 and a3 = -0.064.

Subsequent values of a,, are negligible. Moreover, when al and a3 are combined with the

results of Figure 7, we see that the third harmonic is just a few percent of the first

harmonic. Hence, with very good approximation, we can use the result

where the dimensionless distance is

ii=

- y, +w I I QYb4

L =- L *

(147)

45
10

1

-With image

E

1 1

a

3 0.1 -

L4

.s

2 0.01 -

2

E

0.001 -

0.0001 : 1 1 I I I

0 0.2 0.4 0.6 0.8 1

Dimensionless Distance

Figure 8. Magnetic Field as a Function of Dimensionless Distance.

In the foregoing equation, we have adjusted for the fact that y, is negative. Figure 8 is a

plot of BR (without the image current) as a function of fi .

When the image is taken into account, we must use equation (141), which now

becomes

B,, = B,(-ij) - B,(--fi- L

2x1 . (148)

A plot of equation (148) is also shown in Figure 8. As observed, the magnetic field with

the contribution from the image current reduces the environmental field by more than an

order of magnitude. However, in both cases, the field has decayed roughly 3 orders of

magnitude at one conductor spacing (L) from the source.

46
6. Summary and Conclusions

We have developed a model for determining the space-and-time behavior of the

magnetic field external to a system of moving pulsed conductors. The model includes a

moving finite conductivity conductor in relative close proximity to a stationary conductor.

The analysis indicates that the magnetic field has a space-and-time periodic@ that can be

related to the parameters of the conductors. Furthermore, we present a numerical

example that predicts the magnitude and frequency of the magnetic field as a function of

distance from the stationary conductor. In this analysis we show that the induced currents

are nearly equal in magnitude to those in the source. The eddy currents in the moving

conductor delay the penetration of the magnetic flux. The environmental magnetic field

decays rapidly to negligible levels at distances of interest to the EMC problem.

One of the more interesting and potentially important devices currently in

development is the air-core pulsed alternator. These devices have been used to deliver

large currents for electromagnetic launch systems. The main difference between a

conventional generator and a pulsed alternator is the latter’s extensive use of

compensation. Compensation is a method of reducing the inductance associated with an

alternator’s armature winding. This reduction is achieved by inducing currents in a

moving conductor that is in close proximity to the armature winding of the alternator.

In order to estimate the environmental magnetic field produced by a moving pulsed

conductor, we have assumed a step function current for the current source elements. In

an actual alternator system, these currents would be produced as a result of voltage

generation due to a spatially varying direct current excitation field established in the air

gap between the moving conductor and the armature windings. Of course, this

additional source would contribute to the environmental magnetic field. This excitation

.

source has not been considered in this analysis but can readily be computed using the

techniques developed in this study.

47
INTENTIONALLY LEFT BLANK

48
7.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

.

References

Bennett J., T. Gora, P. Kemmey, and W. Kolkert. “Electromagnetic Braking of a

Metallic Projectile in Flight.” IEEE Transactions on Magnetics, vol. 21, no. 3,

May 1985.

Doyle, M. R., D. Samuel, T. Conway, and R. Klimowski. “Electromagnetic Aircraft

Launch Systems - EMALS.” IEEE Transactions on Magnetics, vol. 31, no.’ 1,

pp. 528 - 533, January 1995.

Summerfield, M. “Development of a Linear Piston-Type Pulse Power Electric

Generator for Powering Electric Guns.” IEEE Transactions on Magnetics, vol. 29,

no. 1, January 1993.

Kilgore, L. A., E. Hill, and C. Flick. “A New Three Million KVA Short Circuit

Generator.” IEEE Transactions (Power Apparatus and Systems), no. 67,

pp. 442 - 445, August 1963.

Pratap, S. and M. Driga. “Compensation in Pulsed Alternators.” IEEE Transactions

on Magnetics, ~0135, no. 1, pp. 372 - 377, January 1999.

Kohlberg, I., C. Le, and A. Zielinski. “Predictions and Observations of Transient

Electric and Magnetic Fields Generated by Electromagnetic Launch Systems.”

Conference Paper, XXVII International Union of Radio Science (URSI) General

Assembly, Lille, France, 28 August - 5 September 1996.

Zielinski, A. E., C. Le, J. Bennett, and I. Kohlberg. “Railgun Electric Fields:

Experiment and Theory.” IEEE Transactions on Magnetics, vol. 31, no. 1, pp. 463,

January 1999.

Zielinski, A. E. “In-bore Magnetic Field Management.” ARL-TR-1914, U.S. Army

Research Laboratory, Aberdeen Proving Ground, MD, March, 1999.

Jordan, E. C. and K. G. Balmain. Electromagnetic Waves and Radiating Systems.

2nd Edition, Prentice-Hall, Inc. Englewood Cliffs, NJ, 1968.

Banos, A. Dipole Radiation in the Presence of a Conducting Half-Space. Pergamon

Press, NY, 1966.

Hill, D. A., and J. R. Wait. “Diffusion of Electromagnetic Pulses Into the Earth

From a Line Source.” IEEE Transactions on Antennas and Propagation, vol. 1,

pp. 145, January 1974.

49
12. Bannister, P. R. “The Image Theory Electromagnetic Fields of a Horizontal Dipole

in the Presence of a Conducting Half Space.” Radio Science, vol. 17, no. 5,

pp. 1095, September - October 1982.

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