close

Вход

Log in using OpenID

embedDownload
NASA
TN D-.4.65
_D
<
<
Z
TECHNICAL NOTE
D-665
THE
By
MAGNETIC
Edmund
FIELD
E.
OF
Callaghan
Lewis
and
Research
Cleveland,
NATIONAL
AERONAUTICS
WASHINGTON
A
FINITE
SOLENOID
Stephen
H.
Maslen
Center
Ohio
AND
SPACE ADMINISTRATION
Oetober
1960
6
NATIONAL
AERONAUTICS
AND
TECHNICAL
THE
By
O
O
MAGNETIC
Edmund
E.
FIELD
Callaghan
SPACE
NOTE
OF A
ADMINISTRATION
D-465
FINITE
and
SOLENOID
Stephen
H. Maslen
!
SUMMARY
The axial and radial
fields
at any point inside
or outside
a finite
solenoid
with infinitely
thin walls are derived.
Solution
of the equations has been obtained
in terms of tabulated
complete
elliptic
integrals.
For the axial field an accurate
approximation
is given in terms of
elementary
functions.
Fields
internal
and external
to the solenoid
are
presented
in graphical
form
for
a wide
variety
of
solenoid
lengths.
INTRODUCTION
!
O
The recent
great
interest
in plasmas
either
as a source
of energy
or as a propulsion
device
has resulted
in a greatly
renewed
interest
in
the magnetic
fields
produced
by various
configurations
of electromagnets.
Of the possible
methods
of plasma
confinement
by far the most promising
appears
to be the use of magnetic
fields
(ref. I).
The calculation
of the fields
generated
by various
electromagnetic
configurations
such as loops_
finite
helical
solenoids,
and infinite
solenoids
has been treated
by the early classical
physicists,
but only
the simplest
cases such as the single
loop have been calculated
for the
entire
field both
inside
and outside
the loop (e.g., ref. 2).
In other
cases
such as the helical
solenoid
or the finite
solenoid
the calculations
have been limited
to the axis (ref. 3).
axis positions
have been done by Foelsch
(ref.
are obtainable
by means
of a large number
which
are valid over restricted
ranges
of
Derivations
4), and the
of the offsolutions
of approximate
expressions
size or position.
The princi-
pal difficulty
in the calculation
of the fields
of nearly
all configurations
has resulted
from the fact that the integral
solution
cannot
be achieved
without
the use of various
elliptic
integrals.
Even though
many of these are tabulated_
the calculations
involved
are laborious.
Such calculations
can, however,
be made using modern
high-speed
computers
since machine
programs
have or can be written
for many of the
elliptic
functions.
The purpose of this report is twofold:
(i) to derive the equations
of the axial and radial field at any point within or outside a finite
solenoid in terms of standard tabulated functions and (2) to plot these
fields for a number of solenoids.
SYMBOLS
A@
magnetic
a
coil
Br,Bz
radial
E
complete
i
current
K
complete
vector
potential
component
in
@-direction
magnetic
induction
component
radius
and
axial
elliptic
in each
integral,
second
kind
filament
elliptic
integral,
first
kind
k
L
coil
n
number
r_ e_ z
cylindrical
ho
Heuman
g
permeability
z_
length
of
turns
per
unit
coll
length
coordinates
lambda
function
L
2
I
qD
DERIVATION
Consider
a solenoid
as
shown
OF E:QUATION3
in the
L/2
_b
z,Z
follewing
sketch:
3
The magnetic
potential
A
field
by
due
to
this
coil
is
given
in terms
of
the
vector
(I)
=VXA
where;
for the geometry
assumed_
Then equation
(i) yields
simply
Br
only
the
A8
component
can
be
nonzero.
= _
(2)
I.
;_Ae
Bz = l _(rAe)
r
For
a single
circular
filament_
Ae
I
;gr
one
has
= 4_i_ifa
c°sR 8 d8
O
!
where
R
is the distance
from the local point
on the filament
to the
field point.
For a solenoid
made up of a series
of
n
filaments
per
unit length_
we have then
<9
L/2
cos
Ae=_/
a_o
V(z
Z)2
+'r
2
8 d8
+ a 2
_
2at
cos
8
or
A0= f /o
2_
where
origin
cos
+ r 2 + a 2
_ = z - Z, _
= z ± L/2; and
Z
to the filament.
On integrating
i o = _
cos e in
2_
8 d8
(3)
d_
+ _2
-
2ar
is the axial
with respect
cos
8
distance
from the
to
_
this becomes
+ r 2 + a 2 _ 2ar cos
__
ae
(4)
0
A more
Ae
convenient
form
= a_nilsin2_ 0 inI_
can
be
found
by
integrating
by
parts:
+
e=0
ar
+r 2 +a 2-2ar
sin?8
cos
de
e)V_2
]_+
+r 2 +a?_
2at
cos
8J{_
4
The first
_2
Ae
r2
+
V_2
use
term vanishes.
a 2
+
+ r 2 + a2
of
=
the
Onmultiplying
_ 2dr
cos
8 - _
_ 2dr
cos
@ -
limits
a2_ nit
eliminates
rearranging
one
term,
terms,
there
and
I
follows
=
_Ao
- _-_- =
_
sin28 + d8r2
8)V_2
cos
can now
equation
Bz, first
as in the
- 2dr
cos
The
0 d0
_ alani
from equation
equation
(5),
_ cos
r 2 + a2
-
2ar
cos
O(r -a
0)_'2
:os
radial
(G)
2_
evaluate
_A@/_r
case of obtazning
(5)
8] __
cos
be easily
obtained.
(Z) and yields
_
-
+ a2
_2 + r 2 + a 2 _ 2dr
_AO_
that
{+
I( a 2 + r 2 - Zar
The two magnetic-field
components
field
is found by differentiating
To get
ceeding
observing
_
2_
Br
the integrand by
cos
The
(4).
is
__
result_
pro-
O)dO
t- r 2 +
a2
-
2dr
cos
__
(7)
If
equations
Bz
_ i
r
_(rA8)
_r
(S)
and
=
2,_
(7)
are
f[
put
into
(2),
th_
result
is
_(a-0)
?cose)de
(r2
+a2 2atcos
+r2+a2 2= cos0]_+
(8)
Equations
(6)
and
(8) describe
the
magnetic
field
due
to
a finite
sole-
noid.
Numerical
results
can readily
be found by integrating
these equations
on a computer.
However,
the results
eal also be expressed
in
terms
of standard
elliptic
integrals_
which
as
already
tabulated.
This
we proceed
to do.
412.01
Consider
(noting
B r.
the
This can be
special
case
evaluated
by Ise of formulas
of
_2 = k 2) )f reference
5.
291.05
and
One has
successively,
Br
=
m
__r
"
+
m_.
(r +
a) 2
IK(k)
m_._._2)sn2u di:+
1(.2- - k2sn2u
_
I
co
O
O
5
+ (r + a)z
i -
K(k)
Br
O
o
ob
!
=
_
Tr
E(k)][+
2k
-
(9)
k J__
where
k2
=
4ar
(a + r) 2
_Z +
In
a similar
integrals.
First
then use formulas
successively
manner,
change
233.19
B z (eq.
the
and
(8))
can
(10)
be
reduced
variable
of integration
413.06
of reference
5.
to
standard
elliptic
to
t = cos @,
There
follows
and
9
Bz __
Bz
r22+
- t
a2
(1 - t _) _2 +2arr2
1
_ni
- 2_(a + r)
_k
Sz =
F
i
(a
_ [._K(k)
+ r
4ar
g_
du
(a + r)2 snZu
[k
- t
sn2u ,
a
0
n__.
+ a 2
-
_-
r)[
+ ]"(a
r)_l
(11)
;%(m,kj__
where
(12)
As
before,
ho(_,k
k
is given
) is tabulated
equation
(I0).
in references
by
5 and
For many purposes_
it is convenient
of the fields
near the axis.
As
r _ 0,
to the following
well-known
expressions:
Br
:
,_
_ni
[(_2
The
Heuman
lambda
function
6.
simply
to
equations
a2,3/_, -
+a2r / . ._]_+
know the variation
(9) and (ii) reduce
(13)
(1A)
_2+a2
A convenient
to i percent
approximation
for
in this range,
is
Bz =
m(l
+
Bz,
2k')
valid
+
__
whenever
4
r _< a
and
accurate
*
C
C
I
CO
0
0
where
actly
m = (i- k')/(l + k'), k' =_to equation
(14) at the axis.
k 2.
Equation
(15)
ex-
reduces
CALCULATIONS
with
Equations
(9) and (ii) are readily
written
in dimensionless
the distances
given
in units of the solenoil
radius.
Then
tions
(9) to (14)
_+
throughout.
Plots
of
-4Br/_ni , are
still
hold
the dimensionless
given in figures
made for the ratio of toll
Note that in figures
1 and
the coil radius
(r/a) and,
coil half-length
(2z/L).
but
with
ij r/a,
_h/a
replacing
form
equaa,
axial and radial
fields_
+4Bz/_ni _
1 and 2, respectively.
Calculations
length
to radius
in the range from i to
2 the radial
distance
is given in terms
the axial distance
is given in terms of
r,
were
25.
of
the
Discussion
The figures
clearly
show that increasing
solenoid
length
decreases
the radial
variation
of the axial field.
This r_sult
is expected
since
an infinitely
long solenoid
has a uniform
field
;hroughout.
For short
solenoid
lengths
(fig. l(a)),
the axial field
increases
rapidly
from
the center
to the wall for positions
near the ee:Iter of the solenoid.
In fact,
at the center
the curve approaches
very closely
that for a
simple
loop.
It should be noted that the radial
field
is always
infinite
2z/L = 1. O and
r/a = 1.
This point
corresponds
to the edge of
current
sheet and would be expected
to produce
s lch a result.
In general,
uniform
fields
with total variations
be achieved
over as much as 60 percent
of the internal
solenoid
if the length
is 25 radii
or greater.
at
the
of i percent
can
volume
of the
I
The calculations presented herein are limited to solenoids with infinitely
thin walls but the results can readily be used to find approximate solutions for various shaped solenoids of finite thickness with almost any current distribution.
Since superposition principles apply, it
is only necessary to approximate any odd-shaped solenoid by a number of
thin-walled solenoids and add the fields resulting from each. The accuracy of the answer is, of course, dependent on the number of separate
solenoidal rings used to approximate the actual shape.
oo
It is interesting to note that the results obtained herein for magnetic fields are closely related to the velocity fields produced by a
lifting
helicopter rotor (e.g., ref. 7). The physical model is the same
but the detailed methods of solution are widely different.
Lewis Research Center
National Aeronautics and SpaceAdministration
Cleveland, Ohio, May 23, 1960
REFERENCES
!. Bishop, AmasaS.: Project Sherwood- The U.S. Program in Controlled
Fusion. Addison-Wesley Pub., 1958.
2. Scott, William T.: The Physics of Electricity
Wiley & Sons, Inc., 1959.
and Magnetism. John
3. Mapother, Dillon E., and Snyder, JamesN.: The Axial Variation of
the Magnetic Field in Solenoids of Finite Thickness. Tech. Rep. 5,
Univ. Iii.,
Nov. 16, 1954. (Contract DA-II-O22-ORd-992.)
4. Foelsch, Kuno: Magnetfeld und Induktivitaet
einer zylindrischen
Spule. Archly f. Elektrotech., Bd. XXX, Heft 3, Mar. i0, 1936,
pp. 139-157.
5. Byrd, Paul F., and Friedman, Morris D.: Handbookof Elliptic
Integrals for Engineers and Physicists.
Springer-Verlag (Berlin), 1954.
6. Heuman,Carl: Tables of Complete Elliptic
Phys., vol. 20, Apr. 1941, pp. 127-207.
Integrals.
,our. Math.
7. Castles, Walter, Jr., and DeLeeuw,Jacob Henri: The Normal Component
of the Induced Velocity in the Vicinity of a Lifting Rotor and Some
Examples of Its Application.
NACARep. 1184, 1954.
8
GI
K_
iiii i__+_:__$i_-t
_J
I
CO
o
0
/,.,
,
U!Y
.,
:x
-_
_B:Li2tj£i
_p
d
c'
®
ii!! i_!l-;._:i
;:
.... I!i_: !: i. I:'
,
]!il]iii_t_:
_T£
_co
"
2:4i ilii i U :'4_2.
T
,. ........_
I:,H
_H
_" , '__
._..,:z!_u:
....
@
4_
-__:ia!2"i,.'_.'''÷_'-:--
o
JR7 ....
•
÷
@
rq
@
C
c6
;:. p v ::W.::
_..::{i
!L' _:;_;:
!:_I":I'G.::
i;;: t;:T:{!!.d£!:T_!_F!_
:_t!i:_,_ 'h,J-,_"_111I_:
r,
_
"
@
0
__';
_
....
! .....
...................t!l
:
imq:,
'
:::ti :: Y: ;: IT":i
__
__
::::t1
44_
;i'::!:
,_'
:_ :
': ....
I_i!//,tlH
_! I
,+....
[ ,
,l
_
_2
......+"
IWir/iiti::}I:dl!:lvtN_ i_,td: ::i_t
i
'
_:: : :
:
'
:
'
'
'_
I:::';_
'
'
.":--.i,}" ::l:-'J2_
: ]:: -Ti4:,_':'
:
'
_ :
"
":',|:.*_t,,
'_:"
....
!
]
I_
_ ,li_
_-
t ....,
2+'
-,m
'i !t';_:Ti,_!, :,_: ,::
iT'-
''
_
_ ,
'
i
:
:
,
:Hh,I
"
,Y_!!!:!::qt
,_
_
:t
:
i :: ::_':._
E
I
": :::
4
£
b_
F:iH:::ili!.hhH:h!i:-fl,:_iu_FtU:-H,:.
,
:"
:
_.,
¢1
,.
'1
;
.......
_
:;t
t
-t:':;t
,}
i:. : : ;!_:
_
''2}
%..
,
,,,
!. Yti:
,
_
1! :!i
.............
'
:
"_:f
:
I_
$1
t:!!:
'
::_'_,,,I,
[ : !:_f:
O
'i[ :
'
co , : i
r"
T .........
r ]
i :1:
[ ................
i! ..........
I ;
:t
ti :
rI_1 .....
_
""
' '
-
"_f
:Tf
q
; '
0
CO
_
"_
C_!
0
cO
LiD
_
Oa
0
bQ
C,.I
Cq
O_
O,1
O1
H
H
r--I
H
_1
:
........
! :i
_Lt_ " _..........
t r_
[ :2'I!_E
,
, ;TH_IFH!I!:I.:'X
_
[:i= '_!.
t
' ;
a3
i'!2
f "_.......
.....
t .....
_:i
i :i:
!
H 4'_
CO
' t
'
......
_
_
:i:
lt'-',l
'
04
'
Or-< I
o
©
oh
!
#q
,g
[a
_H
c_
0
I
o"
oJ
V
II
c_
Q)
A
6)
,.-i
rj
I
H
0
I0
I
<D
0
0
0
•
u/a
"snTp_
°
s¢_IUOTSU_G
11
0
0
Cb
I
'K1
gl
_d
0
,q
!
4
_
0
©
o)
£7
O
r,m
I
©
.H
Cr_
12
!
LO
0
0
0
•
_/_
C_n_P_
sgaIUO_UemIC
°
15
0
0
C_
I
I
0
0
0
0
•
•
I
IS
0
0
0
G)
I
M
0
16
:4;
ffl
_F
F_
o
a,+f
J_t
!;4
.i;i
2.8
.4
Dimensionless
radial
(a)
Figure
2.
-
Dimensionless
L/a
radial
field,
=
4J r/_ni
i.
field
of
a
finite
solenoid.
!
r.o
o
o
17
.4
.8
1.2
Dimensionless
1.6
radial
(b)
Figure
2.
-
Continued.
Dimensionless
"_/a
2.0
field,
=
2.4
2.8
4Br/#ni
2.
radial
field
of
a
finite
solenoid.
3.2
18
I
<0
0
0
.4
.8
1.2
Dimensionless
1.6
radial
2.8
2.4
2.(,
field,
4Br, _ni
(o) L/_ = _.
Figure
2.
-
Continued.
Dimensionless
radial
field
of
a
finite
solenoid.
3.2
19
0
0
O_
I
2.8
2.6
1o
.4
.8
1.2
1.6
Dimensionless
radial
2.0
field,
2.4
2.8
4Br/_ni
(d) L/_= 4.
Figure
2.
- Continued.
Dimensionless
radial
field
of
a
finite
solenoid.
3.2
2O
2.8
!
QO
(D
CD
2.6
2.
2.
2. C
1.8
l.E
1.4
i.
i.(
.4
.8
1.2
Dimensionless
I.G
radial
2.0
field_
2.4
2.8
iBr/_ni
(e)L/a = _.
Figure
_,
- Continued.
Dimensionless
radial
field
of
a
finite
solenoid.
3.2
21
3.0
0
0
2.8
4
2.6
2.4
2.2
2.0
1.8
%
1.6
°H
n
"_
n
1.2
1.0
.8
.6
.&
.2
(f) L/a = lO.
Fi_e
2.
-
Continued.
Dimensionless
radial
field
of
a
finite
solenoid.
22
II
:!!!
i!tt
itH
IUt
I,w
i;i!
o_
M_
lift
A
_r
i!
If'
._r
.4
.8
1.2
1.6
Dimensionless
(g)
Figure
2.
-
Continued.
2.0
radial
Dimensionless
L/a
field,
=
2.8
2.4
Z Br/_ni
15,
radial
field
of
a
finite
solenoid.
5.2
!
t..O
0
0
23
.4
.8
1.2
Dimensionless
Figure
hASA
- L.cFloy
}ielcl,
Va.
_-,_<.
2.
-
Concluded.
1.6
radial
Dimensionless
2.0
field,
radial
2.4
2.8
ABr/_ni
field
of
a
finite
solenoid.
3.2
1/--pages
Пожаловаться на содержимое документа