Download ELECTROMAGNETIC FIELD OF A MOVING WIRE CARRYING

ELECTROMAGNETIC FIELD OF A MOVING WIRE
CARRYING CURRENT PARALLEL TO AN INFINITE
SI,AB *
BY
I. R. CIRIC
538.51 : 621.3.013
The analysis presented relates to a rectHinear wire carrying a.r. parallel to the plane surface of an infinite conducting slab in relative motion to each other with constant
velocity and to quasi-stationary electromagnetic field. The vector potential, the losses
in the slab and the force on the '"'·ire are determined.
For the particular case of d.c., the simplified expressions of the losses and force are obtained by introdul'ing an "equivalent penetration depth" by analogy ·with that corresponding to the pronounced skin effect. The braking force increases initially \\·ith the velocity. presents a 1naxtn111m and decreases artcrw<.'rds to zero for increasing velocities.
1. 1:\"TRODl"CTJON
The study of quasi-sfatiormry ekctromagnetic field in the presence
of moving solid conductors is important fort.he detumimi,tion of losses, forces and torques in the systems of electromagnetic brnking, of electromagnetic levitation of moving conductors, in solid armatures of electrical machines or conditioned by moving wires carrying emTent in solid conductors.
'Ve begin by considering an infinitely long rectilinear filiform conductor carrying a.c. (1) parallel to the plane surface of ~• semi-infinite region, linear, homogeneous and isotropic, of permeability µ and conductivity er, with the relative velocity v betweEn wire and slab assumed to be
constant. The medium outside the slab is nonconducting, of permeability µ 0
i =I
f2
sin (wt+ y,).
(1)
2. :FOR\IULATION OF THE PROBLElI. \"ECTOR POTENTIAL EXPRESSIOXS
Choosing the coordinate system shown in Fig. 1, the vector potential
of the magnetic field A has only z-component,
A(x, JI, t) = k.A(x, J/, t),
(2)
and depends only on the coordinates x and y, the magnetic field having a
plane-parallel structure,
1
H=-v XA.
µ
* Dedicated to professor Rernus R3.dulet, mcn1ber of the Academy.
Rev. Roum. Sci. Techn. -Electrotechn. et Energ., 2-4, 3, p. 375-388, 8ucarest, 1979
(3)
376
I. R. CffillC
2
The vector potential (2)-(3) may be determined by using either the
coordinate system fixed to the slab, in which the wire has the velocity
-iv, or the coordinate system fixed to the wire, in which the slab has the
velocity fo.
y
!DI
--=- x;.:l;):--
------~-
IIJ
iy
>'•~
I'.
~vv
Fig. 1. - Rectilinear wire
and infinite slab in relative motion.
z
2.1. VECTOR POTE:>1TIAL SOI.CTION IX THE SLAB COORDINATE SYSTEM
The vector potential equations in this case arc [l]
oA
ilA - µa--·= 0
·.11 <
at
ilAa
=
'
µ 0 i3 (x -
-
'
a: 0
+ vt) ';;(y
-
y 0 ),
y
o,
> O,
(4)
(5)
whne delta functions [2] are introduced to define the current density
outside the slab, x 0 and J/o are the initial coordinates of the wire and the
constant electric field determinfd by the scalar potential has been neglected, with the continuity conditions at all points on the surface of the slab,
i.e. at y = O:
(6)
_1
_04 i
µ
O!f lv~o
__
1 i}_Aa I
~'o
(7)
oy lv~o
The above equations may he written under the equivalent form:
&A
aA - ua--
·
at
= 0
'
ilAu = O,
with the continuity conditions at JI
=
A lv~o
1
µ
< o,
0
<Y<
<y<
(8)
Yo1
oo,
Yo
0 and y =Yo:
l
,-, . i
:u~o
(9)
(10)
(11)
A, lv~O•
oA:
oy
=
y
0A 1
==--fl 0 i)y
Y=O
(12)
'
(13)
I
( iJA,
-.- -
µo
oy
r1Au ) !
•
j
= i';;(x oy
) ~y,
- . - .-
x0
+ vt),
(14)
377
,FIELD OF A MOVING WIRE PARALLEL TO A SLAB
3
in which the current i has been assumed to be distributed at y = y 0 under
the form of a current sheet of linear density i~(x - x 0 + v t).
Expanding the delta function into Fourier integral [3] gives
x. + vt) =I V2 sin (wt + y
id(x -
1)
_!_ (""cos/... (x
7t
Jo
IV3! (''°{sin [( f..v + w) t + y, + f..(x-x 0 )]
=
2n
+ /. . (x
-
Jo
-
- x0
+ vt) di.. =
sin [( f..v - w) t-y,
+
(1.'i)
x 0 )]} clA.
Superposing the effects of the elementary harmonics of angular
frequency f..v ± w yields the vector potential expressimrn (the wire current is assumed to be returned through thP ~lab):
y.;;;
A,
=
o,
l
J:olV2 Im("" e-'-Yo (e'-Y + -~---:!3L
4it
)o
/..
1 + !3x
e-J.y) eil(i.VTw)
•+Y;I -
(17)
_ (e•Y
+
1
1
A
=
rv2
µ
__()_ _ _
00
e-'·t[( e•Yo + - __
~+
,,_._
+ !3x
Im ~ _._
47t
II
!3i: e-'-Y) cill1'•-wJ •-Yil] cJ'-1•-'oi d A
'
-
+ !3i:
"
1 -
1.
e-i.y, )
eJ([l.v+w) l+Y;) -
1
(18)
where
j=V-i.
(19)
2.2. VECTOH POTENTIAL SOLCTJON !'.'!THE WIRE COORDINATE SYSTEM
Here, the equations satisfied by the vector potential are either
~A
aA
aA
ax
dt
- µav-.- - µa·-.-
=
O,
y
< o,
( 4')
y
> o,
(5')
378
I. R. ClRllC
4
with the same conditions (6)-(7) at y = O, or
8A
AA - µav -
ax
-
iJA
µa ---
0
=
&t
'
AAr = O,
0
y
< o,
(8')
<
Y <Yo,
(9')
Yo< Y
with the eontinuity conditions at y
=
O and y
=
<
oo,
(10')
Yo
(11')
_I___
aA I
µ
oy u-o
=
~ oAr I
µo
&y
(12')
u-o'
(13')
.:!:_ ( a_Ar - iJAu ) I
µo
By
oy
= ia(x - Xo)·
•-t.
(14')
In this coordinate system, the vector potential along with all field
components will have a sinusoidal time variation, like the current in Eq.
(14'). Looking for a solution of angular frequt>ncy "'• which has a complex
representation of the form
A(x, y) =
~="" A 0 (y) eJ1.(x-x,1 di.,
A(x, y) =Im
{V3l ei"'' !(.11, y)},
(20)
(21)
and taking into account the expansion [3]
(22)
finally yields the vector potential expressions:
y.;;;.o, (23)
A,=
-
11 [
~o,__
41t
~oo e-AYo
- [.( ().y
i.
o
1 - !'r>.+
+ ___
_•_ e-•Y )
1 + f3x
eiA(•-•ol
+
(24)
+ (e•• +
1 -
1
f3i:*
+ f3r*
e-1.y) e,-i•t•- ..1] di.
'
FIELD OF A MOVING WIRE PARALLEL TO A ·SLAB
Au = ~[ ("'
c-).y
4ri: Jo
+
(c'Y• +
[(e1.,, + 1 -
/..
1
1
1
13A"* e-1.y,)
-
+ 13>:*
M
+ 13t
379
c-'Y•) ej'l•-•ol +
e-il.l•-•ol]
d"A,
Yo
<. y <
oo,
(25)
where the asterisk signifies the complex conjugate of the respective value.
The vector potentials (23), (24), (25) are identical to those in (16),
(17) and (18) respectively, if one considers the transformation (21) and the
substitution x-+x + vt when passing from the wire coordinate system to
the slab coordinate system.
2.3. PAllTICULAR CASES
a) l!'or v
=
O, using- notations (19) correspondingly,
R + - R-* ·- µO k1., - A
f"A -t-'A
--=f"A.w'
µ
}_
(19')
gives the known expl't'ssions of the vector potential of a fixed rectilinear
wire carrying a.c. in the prcsrnce of an infinite conducting slab:
A
µ I ~"' p,,
= ....Q=-.
-
c1'""' •
----cos/.. (x - x 0 ) d"A,
o /.. 1
;c
J~ex>e-1.Yo(
= ....Q=-.
- - e•Y
µ
2ri: o
y..;o,
+ 131..,
/..
+ 1-Al'l., e- 1Y)
1 -l-131.,
(23')
cos/.. (x-x 0 ) d"A,
(24')
I ~"' -e-•y
An = µ.Jl-=..
- - ( e1Yo
2ri: o /..
+
1-
1
-13,~ e-1.y, ) cos/.. (x-x 0 ) d/..
+ 131.cu
'
Yo<.y<oo.
(25')
b) For w
kt
=
=
k>:
=
O, using the same notations (19) correspondingly,
Vt.. + j/..vµa = kl.••
2
13t
=
13>:
=
£Q.. k'N" = 13,,,, (19")
µ
/..
gives the expressions of the vector potential of a rectilinear wire carrying
d.c. in the presence of an infinite conducting slab, with the relative velocity v between them (in the wire coordinate system) :
A
=
µ
T
__
<>"-_o_
7'
oo c-Ay0
Re ( -·-Jo
A 1
A 1 ~- _11-ol_o Re(" e-1.y,
2ri:
Jo
Arr= µolo Re("'
2ri:
Jo
where
3 - c. 1437
/..
ekAvll
- pi1.l•-•,I d/..
+ 13Av
(e•Y + !-=-~~ e-l.y) ei•l•-•ol dt..
1 -!- 13,,
y<.O,
'
e-AY (e"Y•-j- 1 - 13_'.! e-1.r,) et"A(•-•ol di.,
/..
1 + 13,,
10 =IV'§ sin y1 is the d.c. of the wire.
(23")
y 0 <.y<oo,
(25")
I. R. CllU!C
380
3, ELECTRIC AND MAGNETIC FIELDS
The electric and magnetic field strengths (always defined in the local
referential, attached to the moving bodies) can be calculated by using the
vector potential either in the slab coordinate system (16), (17), (18)
iJA
1
H=-V
E=E.=--,
iJt
- E
E 1.11=
•1.11 -
µ
iJA1,n
-
1
X(kA)=-VAxk,
H1,II = -
~'
µ
1
µo
y..;;o,
VA1.II x k,
(26)
(27)
or in the wire coordinate Rystem (23), (24), (25)
E =: E
-·
= - JW
0
1
A - v a.A
-
E II= l!J.1, II=: -•I
'
H= -V,4 X k,
ax'
. A
y:s;;o,
(28)
µ
JW~,11 - 'V
iJA1 II
-iJ ·.- '
(29)
IL'
where the scalar potential component of electric field is neglected.
For example, Eqs. (28) give
E
-
=-•
E =
µ I ~oo e-1'y,
_ J. ~
-27t 0
/..
[
p,v
"i:Y eiA(•-•ol _
+ w) _e~1
+
~t
(30)
(31)
etc., for y < O•
.The electric and magnetic fields (28), (29) are identical with those
in (26) and (27) respectively, if one takes into account the transformation
(21) and the substitution x-+x + vt when passing from (28), (29) to (26),
(27) correspondingly.
4. LOSSES IN TUE SLAB AN.D FORCE ON THE WIRE
4.1. LOSSES IN THE SLAB
The evaluation of the losses in the conducting slab can be done
with the aid of Poynting's vector. The electromagnetic power received on
the length l along the z axis of the slab is
p = _
z,.., ~ x "(l I·~ dx.
l_oo
r-o
(32)
381
FIELD OF A MOVING Wiru: PARALLEL TO A SLAB
7
The mean value of these losses is the real part P J of the following
integral [1]
(33)
Elementary calculation gives :
µ 0 12 ~oo e-21.y, [
PJ= - l --'Avlm
27t o 'A
1
1
+ 1 + (3r }+wlm{ 1 + (3t
Q = l µol2
-- ~
00
e-21.y,
- - [ l.v Re {
27t o A
I
1
+
1 + (3t
1
- 1 + f3r }]d'A
{
(3t - - ----'--(3x
}
2
2
(3x 1
Il
(3f 1
1+
+
(34)
+
(35)
+wRe{
(3x
[l
+ (3t
2
+
[3J:"
[
1
1
+ (3f
2
}]d'A.
1
4.2. FORCE ON THE WIRE
Using the elementary expression of the La place force
(36)
for the mean value of the force on the length l of the wire we have
where .bexi is the vector potential of the magnetic field conditioned only
by the currents induced in the slab,
=
(38)
µ 0-J
j""
e-1.(1,+Yl
47t"
o
>.
[
1
1
-
+
r;+
1-'X eil.(•-•ol
(3t
+
1
1
- f.l-•
I-'~
+ (3i;""'
e-il-1•-z,J ]
dA.
382
I. R. CIRllC
8
Therefore the two components of the force on the wire are
F mx
= l Re
{J. * a;·~·-}
lx~x, =
X
Y=:Vo
- z l'oI_" (oo e-21-y, Jm {
2n Jo
1
1
8A · · }[ x~x,
F..,, -_ l Re { ] *.~ext_
-uy
y~y,
+ ~t
+ __l ___ }
1
+ ~~
(39)
dA
'
~
-
(40)
It is easy to see that the power F mx v, neces8ary to as8m·e the relative
displacement wire-slab, iR that part of losses in the slab which remain8
other than zero for w = 0.
4.3. PAHTICL:LAR CASES
a) For v = O, using relations (19'), we have the known expressionH
of the losses [ 4) and force corresponding to a fixed rectilinear wire carrying
a.c. in the presence of an infinite conducting slab :
(34')
(35')
(39')
(40')
b) For w ~ O, using the notations (19") gives the expression8 of
the losses and force in the case of a rectilinear wire carrying d.c. of intensity 1 0 in the presence of an infinite conducting slab, with the constant
relative velocity v between them :
PJ
=
-
{1}
0
l µo J2oo
v( e- 2'Y•lm - - - 7r
J.
1 + ~ ••
dA
9
FIELD OF A MOV!NG WIRE PARALLEL TO A SLAB
Q
=
o,
383
(35")
(39")
Pm
"
= - l _µol~ ("" e-21,y, 1 - I~" 12 dA.
2n Jo
11 + ~•• 12
(40")
In this case the power necessary to assure the relative displacement
between the wire and slab is equal to the losses in the slab,
I.l
PJ = Pm,V
(41)
c___ _ _ _ _ _
5. SUIPLER EXPRESSIONS OF THE LOSSES AXD l<'OR!;E
The integrals in the losses and force expressions can be evaluat.ed
under certain conditions in order to obtain simpler expressions from practical stand point.
For imtancr, from (34"), (39") and (41), we can write (with (19"))
(42)
where
• =
00 -
When
"-u
<
2
C~:
r
=
V-2Yo
vµcr
.1/oVµcr,
•
(43)
(44)
we may expand the radical in (42) into series and retain a convenient number of terms.
For values which do not correspond to this condition, i.e.
A0
if
2
>
2
(Yo
/)
0
r
C~:r > 1
'
(45)
384
10
I. R. cmDC
the integrand iW,pidly decreasing with ).0 , and we may consider that the
above integral is mainly determined by the integral from 0 to oo in which
the radical has been expanded into series. We obtain
(46)
+
[1 _..!:.. (~)"
Yo
82
4
+ ..!:.. ( ao )
32
2
4
84 -
••• ]
Yo
where
(47)
Retaining only the first two terms in the radical series ;}C:w.6
PJ
=
JJ',.,v
= -
·µ 0 Jg
7t
.!.. Im{!
Yo
(co
a Jo
e-
2 •'
ds (48)
2 [ O ~co e-2•' ds+n( co e-2•' ds,
]}
-a2
o s - s1
Jo s - 82
where
ao
1 -fLo- - (1 -3,
")
a-=
4 µ Yo
(49)
FIELD OF A MOVING WIRE PARALLEL TO A SLAB
11
If Is1 , 2 I > 1, expanding
five terme ~C:et'.l,,
1
8 -
385
into series and retaining the first
812
·
13
1
[l - 2 ( ILµo) ( Y o• ) +
n
PJ=FmxV=l--et•
21tYo a
+ :: (( :0
r- ~ )(:: rJ
(50)
which has sense for
13.
!Lo Yo
1
µ
(51)
--<'
where
y; :0.
=
=
For a nonferromagnetic slab (µ
PJ
= Fmx 'V = l ___!L ~
21ty 0
a
[1 -
(52)
µ 0 ) expression (50) becomes
2_
2
(~) +
Yo
13 2
S7t ( • ) ] ,
128 Yo
(53)
where
(52')
<
In the case 13.
y 0 , the losses and the braking force can be calculated with the very simple expression
16 IX•
PJ= F "'x'V=l---,
21tYo a
(54)
identical -f.,>t ~I!<. losses to that in the case of pronounced skin effect of a
fixed infinitely long rectilinear wire carrying a.c. in the presence o:f an
infinite conducting slab, if
IX.
where 8 = _!_
IX
slab [5].
=
V
2
wµ 0 a
=
IX
=
v
W~oa,
(55)
is the corresponding penetration depth in the
386
12
I. '.R. ClRJJC
It is useful to give the differences between the approximate values
( 54 ), (53) and the exact values (46) of the losses or braking force for different values of the ratio 110 • The exact value (46) for a nonferromagnetic
ilo
slab can be written under the form
(56)
where
(57)
Of being the integral in (46) for µ = µ 0 •
With the first correction in (53), the exact value (46) is
where
(59)
and with the second correction in (53)
PJ =Fm,v
=
z_ll __ ava [1- ~ (a.)+~
(Yoil")
Yo
2rcy 0
2
128
2
]
k',:
(Yo),
il0
(60)
where
(61)
Numerical results for the coefficients k'P, k;, k'; and the corresponding errors when using the Himpler expression (53) with one, two and three
terms respectively are given in Table 1 and k 11 is graphically represented
in Fig. 2.
FIELD OF A MOVING WIRE PARALLEL TO A SLAB
13
387
Table 1
Valueg of c:oeffleleats k1,, A~, A:~ and eorrespon411ng error• a:, a', e"
1
•[%]
1
2
3
5
10
20
0.4026
0.6494
0.7552
0.8480
0.9221
0.9606
k'
•'[%]
k"
JJ
<" [ %]
1.9919
1.0801
1.0288
1.0090
1.0021
1.0005
-49.8
-7.44
-2.80
-0.89
-0.21
-0.05
1.0333
1.0023
1.0004
1.00009
1.00001
-3.22
-0.23
-0.04
-0.009
-0.001
"
+ 148.0
+ 54.0
+ 32.4
+ 17.9
+ 8.45
+ 4.10
I
I
1
o..,
Fig. 2. - k., variation with
ratio _!!__o_.
ilo
v
1
o.Y
0.2
0
5
10 15 20 25
f;
II. CONCLUSIONS
The electromagnetic field of an infinitely-long rectilinear wire carrying a.c. or d.c., parallel to the plane surface of a semi-infinite conductor,
for a constant relative velocity between them, has been determined either
in the slab or in the wire coordinate system, by solving Jihe equations satisfied by the vector potential of the magnetic field.
In the slab coordinate system each field component is a superJK>sition of elementary harmonics of continuously variable frequency, and
in the wire coordinate system the field components have a sinusoidal time
variation (if the wire is carrying a sinusoidal current) or are constant in
time (if the wire is carrying d.c.).
The exact expressions for the losses in the slab and for the force on
the wire have been deduced, and in the case y 0 vµcr > 1 (relation (45)), these
expressions take the simple form of (50), (53). For a nonferromagnetic slab
the limits of validity of these very simple expressions have been established.
When Yvvµcr ~ 1 (i.e. Ii,~ y 0 ) the corresJK>nding expressions of
losses (54) (as well as (53) with the first correction) is fonnally identical
to that in the case of the pronounced skin effect, where Ii, =
.!..
(relation
(52')) is an "equivalent penetration depth" for the d.c. case, "'•
the angular
frequency of the equivalent fixed sinusoidal current determining the sru:ne
losses being (relation (55))
(62)
[. B. C1lUIC
388
For instance, at v
14
= 16 ~ , the corresponding frequency is f. = :
i.e. lHz when Yo = lm, lOHz when Yo = 10 cm, 50Hz when Yo
,
0
=
2 cm,
etc. At the same y 0 , the ratio -"'- must be constant to obtain the same
J.
losses in the slab.
For high values of y 0v µ 0 cr, i.e. of Yo the magnetic field becomes
a.'
practically tangential near the moving solid conductor surface and the
outside field can be determined independently of the extinguishing inner
electromagnetic field. As one can notice, in magnetohydrodynamics
y 0 vµ 0 cr represents the magnetic Reynolds number [6] and for great Reynolds number8 the magnetic field line "freezing" is predominant on the
magnetic field line diffusion.
An useful remark is that at a given distance y 0 , the braking force
increases initially with velocity, the maximum values being reached
approximately within the range ll 0 ::::: y 0 , and decreases afterwards to zero
for increasing values of velocity (relation (60)).
Receioed Marci•
u.
1919
Polylechnical Institute Bucharest
REFERENCES
1. R. RAouLET, Razele teoretice ale electrotelinicii, vol. 4, Bucharest, 1956.
2. PH. M. MonsE and H. FESHBACH, Methods of tlieoretical physics, Part I, Chapt. 7, McGraw-Hill
Book Co., Inc., New York, Toronto, London, 1\153.
3. D. lVANENKO and A. Soxowv, Teoria clasicd a clmpului, Chapt. I, Edit. Tehnici'l, Bucharest,
1955.
4. A. TIMOTIN, A. "fuouLEA and C. FLUERA~u, Ctmpul electromagnetic cvasistafionar al unor fire
reclilinii parcurse de curent alternaliv tn prezenta unui ecran plan infinit de grosime
(initd, St. cerc. energ. e\ectr., 16, 3, p. 481-501, 1966.
5. C. I. MocANU, Calculul adlncimii de piitrundere a clmpului electromagnetic In condactoare, St.
cerc. energ. e\ectr., 18, 3, p. 637-660, 1968.
6. J. A. SHERCLIFF, A textbook of magnetohydrodynamics, Chapt. 3, Pergamon Press, 1965.