SKIN EFFECT RESISTANCE RATIO OF A
INS'.
22 OCT 1929
CIRCULAR LOOP OF WIRE,
LIBRAR'(
A THESIS
by
VICTOR J.
DECORTE.
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENT FOR THE DEGREE OF
MASTER OF SCIENCE
IN
ELECTRICAL ENGINEERING
FROM THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY.
Signature of Author
Certification of the
Department of
Electrical Engineerin
ACKNOWLEDGMENT
The author wishes to extend his most sincere
thanks to Professor H.
B.
Dwight of the MassachusettS
Institute of Technology in appreciation of his many
suggestions and of the assistance he rendered in
checking the various parts of the work done in
preparation of this thesis.
V.
J.
D.
Cambridge, Mass.
M ay, 1929.
Professor A. L. Merrill
Secretary of the Faculty
Mass.
Inst. Technology.
Dear Sir:
-
In accordance with the requirements for the
degree of M 'aster of Science, I herewith submit a
thesis entitled: " SKIN EFFECT RESISTANCE RATIO OF
A CIRCULAR LOOP OF WIRE."
Respecfully submitted,
166~
5 03
CONTENTS.
I. Derivation of Formulas.............................1-3
Current Distribution...............................21
TotalCurn...................2
Power o .. . . . . . . . . . . . . . . . . . .
.2
Total Current....... .............................
.30
Alternating Currwnt Resistance..............,...*.30
Skin Effect Resistance Ratio......................32
II.
Numerical Examples............................34-38
III. Analysis of Examples...........................9-41
IV.* Conslus ions................................
V.
Appendix.......,
... 0
..........
a
*9......
. . . .
.42-43
..,44-53
1* Some Definite Integrals Encountered in this
44
.. ............
. ...
99
09
9
.
.......
Paper.... ,. 999
2. Some Useful Indefinite Integrals................46
3. Power Loss in a Circular Loop of Wire Carrying
Direct Current
.
. . . ..
. ..
. . . . . . . ..
..
. ..
. . . . ..
4. Power Loss in a Hollow Disk Carrying Direct
CurrentI...............51
.48
DERIVATION OF FORMULAS.
In the study of alternating current
distribution and closely related phenomena such as skin
effect resistance ratio and power losses, in a circular
loop of round wire, the effect of the curvature of the
wire has never been considered. This paper presents
some mathematical expressions which have been derived
with due regard to curvature,
Let us consider one turn of round, nonmagnetic, homogen&ous wire carrying an alternating
current of complex value .
Let
C
and of frequency
4
.
be the center of the loop under consideration (Fig.3)
and m, the mean radius of the loop.
The intersection of the loop and a plane
passing thru C and perpendicular to the plane of the loop
is an equipotential surface as all streams of current
2
flowing in the wire strike this surface at right angles
there are no radial components of current (Fig. 2)
i.e,
Let M and 0 be the traces of two filaments of
current on the equipotential surface under consideration.
The difference of the resistance drops at 0 and
M
(meaning the resistance drops of the two filaments passing
thru 0 and M respectively) in this equipotential plane is
2io a
where
,-o
o
-
-
t
- )
_y)
are the current densities at 0
and i
and M
and
respectively,&-is the resistivity of the conductor
c= r siv
and
e
e
(r being the radius of the wire to the point M
the angle between the reference axis and the radius
r ),
If
-
we let i C,,)be the current density at
( any point in the equipotential surface under consideration ),
the total current flowing thru a small element of area
at
P will be je
otSt&.,jand the drop at 0 due to
c
d3Sc.)
osLeM)
at 1' will be
?
where
~Po
Ai(S
.) OSg
lohas the usual meaning of mutual inductance of the
two circular filaments thru 'P and 0 .
A *Y.
z
setting up the expression-for the difference
In
in
resistance drops at 0 and M , it
was considered that the
wire is made up of an infinite number of small eircular
coaxial filaments - this assumption is justified since the
current undoubtedly flows along such stream lines. On the
basis of this assumption, we may express M-po by one of the
many formulas expressing the mutual inductance between two
coaxial circles.
When ( < a
the M of two coaxial circles
(Fig.3 ),
may be expressed by Maxwell "s formula:"
M
(Ci +
[ Am
!-n
=0
In
-.. )
-
(z
p
and
c= C 0
Now the drop in M due to i
if
M
dScc,jat ~P is
expressed by Maxwell s formula
~
7'-c
8(a+c)
+2(atc)
where
--
the case under consideration,
C = St
where,
-
+
c'= Ps-m 6.
P
A
Fg.3.
SFormula 10, Scientific Paper No. 169.
c
.
4
Let us put
and make the following transformation
Se*
-
_
=an
(
+ 477
-++
Cx
L
( L)
7c~G,
(A
+
+,e
C
+
I77Cf
q7 zzS
C,
-i W .
2
+* .-
zaal
+
s _F-'Y
J
Za
(1+
-
*-
(
-._-
(
/
I +
._
.)-/]
fr
-
a.a
-
-
(
, a,
The term
S<+'.
~ - )
('S
L
a1
- -
~
( I +..)~'
(i
i +
A-7
-
jI
a)/
5 L
411
+
(
*+
'
_
e,
- _
2 a,
5
(,+
/
_C.
z (a.+C)
2 CL072
We may now write
))may be expanded into a series,
since
b+..
{ a - b)~= (i-j+b-
and in this case
b'x<
4a
2
is smaller than unity since a, must
-
necessarily be greater than c.in order to satisfy the
condition imposed by Maxwell s formula.
Thus, we have
-
+
0
The first
term only of the logarithmic series
1It
x
-
=
x
- _L!
+*
3
- A-&/ +..
is taken since square terms have been omitted in the mutual
inductance formula. The series is rapidly converging- since
c, is much greater than the largest possible value of C
( in ordinary radio coils, for instance. )
5
our expression for /pp/,
Substituting this in
Y-
=nan Z-
_
terms in
where all
for
/A
7a
2+nC
-ana.7
+n(I
1 74a
-
have been omitted as in
a
we have
the express1i?
ApO1Since the impedancedrops at 0 and M are equal (Fig.2),
we have
2n ac7'
0
-
2n
+
(A9p
/piv)-
'
=S(eJO
A
If we let
7.
the above expression becomes
+'
Acryj =
1(N)
O /.A7
(7
2-o- -t
+
'd
-0
present form, this expression for current
In- its
not useful. We will now set out to put it
distribution is
in
a form which can readily be used for calculation purposes.
We must first
-
o7 --
na
a(77.A
C
i/
-i97a
n
determine the value of
c'
)(.
)
)(+
-I n
j
-
-n
+ q~ a,
z
'
ILa
-Mm.
2 +f
a
z/r
-
f11po
7
s
+
n
3
af
a Ta
2
Combining terms
PO - /M
- i.-n
a?)
It
is
=4
7a
5
(7
+ 47 a {xf
2.
+
-
2a
a
+ l na C _)
2a
not very desirable to have a -I- factor in
a
logarithmic expression because at a certain stage of the
'-
6
work it
wihldbbe necessary to express 5 in terms of a,
r,
and 9 and the trigonometric law of cosines would
probably be the only means of making the
transformation.
The resulting logarithmic expression would be very
cumbersome and considerable trouble might be encountered
in performing the necessary integrations . It
thus
is
preferable to rewrite the above expression so that E will
only appear in the term
P-
=L]
a
S
4
On
(
S.
Thus
qa.
)f
Using polar coordinates (r,O)to indicate the point
A and taking the origin of the system at the point 0
the axis of the loop,(/,
#)will
be variables of integration.
Substituting the above expression for
1Vp - hPM/in
+o
a e,(a
where to
+
kCho(~~
00)
d&n.)
--
we have
e =/su,
and putting
on
+
d o dp=element
c~)~
0,
77
of area
[
a
a,~ 2
rd4
o
-
dS.
This expression forViggray be sirplified
4(r.)
40
&
aT c
4
ys,
1
Z'2nClc
.+.
2
7
It
is
necessary to express
S
in terms of p
,
9
and ; before the integration can be performed.41 admits
of absolutely and uniformely convergent developments,
being the distance between the points(rO)andA
For
<.k
oe/)
).
00
and for 0 :5: e
Fr
The above integral must be solved by means of
Newmann s method of successive approximations.
As a first approximation, let us assume a
current of uniform density in the loop i.e.
Ayr.e)
C9)
o
To get a second approximation fore( ,,,we may
let '
0 in our expression fori'.9P, and perform the
integration.
In order t& simplify the work,
integrals will first
the component
be evolved.
Replacing-t.iby its absolutely and uniformly convergent
development,
all terms depending on (C-9)
will vanihh
during the integration process.
0
Integral equation for skin effect by Manneback,
page 140
8
We thus have
I10
(/
= 2n(
)
d4 d
217
/0
. /L
&L)
r
no
+
-Ad A.
271J-7
o
, we get
).
substituted inj{(-
When the lower limit is
an indeterminate form: 0. 00
21t./J
as A approaches
0
evaluated by L Hospital s rule.
The value of
A -t2-
.-
/
= 4$m
/-
-tm
=
+i-Oo
0
i
e
may be
0
(-..2) r
-
3
o
2
and in general
.np/__= Lm
t~ t
/J+e0
_..imim.
0 becomes
-
Then the term 2-l(
an o
zer
-- '
11
d /i2 = 71
... /
2n
)
(
/ 4tt*
f&].
n
a
n r2
=2-
b c&d
In the solution of this integral, we encounter a
series of integrals of the following type:
277
cc'4 -A (
p
- O)
S-t.v
%
C19
0
2
-
17
IL
Sg
+L
C 0
4
O
I-,cA-)(b
z{ A -)
-p
-o
becomes zero and the aboife
integral gives
2 -rn
/e b
CcI(/++)
+ i&
.J o
2(-A+/)
Jo2
Advanced Calculus by F.S. Woods page 15.
271
9
= o for all values of -A other than I.
For -4 = 1
0<+
- 2.01
'r toln
Cq[ I
277
+~~~
A
4
f2j~~
1177
0
.
77 S-L,
Hence replacing
4-
A by
its absolutely and
uniformly convergent development, we must only consider 'n r
and the cosine terms for
Thus
(~
On
)
-
17
C60
do
rdS+-/9
0
3
oj ( (p - 69)
0
s
0. d4 d 0
0
o
m
t
A2/-,S
277
of
77 S-
7S-i4K9 dO.
dO
0
-77 e 3
/0
-n7
r
io/-vJ7nt
si--
z
2
-n r mto
T
Instead of infegrating terms containing
directly, it is simpler to split up . 4 r
' in the
difference of two logarithms i.e.
.2r-,
n,
and then perform the integrations.
- Z
-e
/.
.,
10
ii.
/0A&
277 /
S 77 n
"i
.
A&/- d4 .
i
n
-
Z'eM ,;,v
171n
2 1
2/
z~J
2
~Y"
/ do do96 =2
Iv.
z7
=
77
z
M.
=77
/
4
All the integrals necessary to obtain the second
approximation have been obtained. We may now write down our
(See eq. 1 )
second approximation for.A(,f
A,(#,O
+
a +a
')
271(a-+t
7-
r'S3-K
*
7
o P
ZqJ
44
+O
-t-
77AoC C
2
2
/
/
*. a r
7,
/
At
z
S776
C
"A
(
2
3.
m-
)
12
Rearranging terms and simplifying
m'S 4Oi
_
-f4
2.
Replac ing
a
C'by rTw',
--&
+
3 S.4'-4 g
c
J
we will have
<2.
P,
q ( a. + MS<Ar
r)~t &-~449
3 C
-S4A.9
4 a.
2a
-- c#.
at +rx. 9
+
I-
,t (Ca+c')
e,.+ c'
S-&$4
S
M7
I Sru 0
+ ;
Jt
2c'-.
Combining terms
yCr. 9)
=
o
a+sn 9
+
+go
a
2
/
+ 3
r +4A
(2j
a2
If
in this expression, we let r=/J and 9 =
we
w,
11
obtain a value of(4.jwhich may be used in determining our
third approximation.
Thus for a third approximation, we may- substitute'
=a
/
+ ,Ao
0 0,a
+ /v :c
+,/o:h @
I
fl2.,QJOb
-~
3/_3
+
in eq. 1 which is
t~'(r.
~J
00.(,,6
4-
-
1g
A
2.77 (a,+e)
2
In order to obtain integrable equations,
=
( -0'
is
We know that
necessary to change the form of/'p.4,,
-j
it
+
-
..
___
Thus
A
=
4{pOp)
o
((i
-
ctv!1)
fr
$*"i~
1-'"
2
+
+l
310 3
+
'+
00[
a-
+3
/V
1
+=
- .,
a.
terms in
neglecting all
.
Collecting terms
44
p
=
.O
A
o
mi
-
_'
s./
_
_
_
@
+
Ao
-
4
3
0
-
12
and may be considered as being
identical to (3)
which is
the second approximation in our process.
Let us denote the different values of
(r6)
obtained by substituting the component-parts of the above
expression fori(p.Ojin the integral equation forA (r.e)by
a'(r.0J t-(b.
Zj (v.t)J
may be written down directlysince we have
yel
from eq.
2.
f
Thus,
already solved for it.
.11
6L .+e'
e __
zs
7.__3S
M~8
''
4 A
2a
4(a+c')
%%
__
-
,
c'*
CJ
The integrals required to determine the value of
A.'r.)will now be worked out.
-nrt
-2"
/7) 3,4%
.Z,
In the solution of this integral, we encounter a
series of Antegrals of the following type;
COS
p -5):.
A0 'A
eos-4 p os11A
0
=
x4't A
-
[ _
p
/c
c<
& 9
++
'-Vd 9dI
+
*, A4~ s&
p* --
Z
-)
$
4
+z
-- zr___A_-_
L(Rii
p
27-7{&
s-
-
#
.s--
A -!+'
'+-"-
{
A+
dczk
A9) :2<
>-
i+A.-,
47
i
110
13
values of A other than Z..
= o f or all
{
If
%t p
Co 2 0 F0#
2, the integral becomes
or 4-
-1)2=/
?
-
/
0
2
~f~f4f~j
L
Ti
10
+
12
6
~2
T?1
=-
((2r)
Thus
S3z
4
7
--
0
0
- 1
W, 2 ;ig
Ma) I II33
277
cso
I
3
/
/
I
-fo
g
277
p-
0
7" 277
r~7/0
2-T7
c -s2 (
p - eq)
* 2773
70A
X42
p
3
ZZ,
z
_L
A
doo d1#
_
0
z
0
l
'
i
--.
c22(pZ
-I
d2
I-
(
- 0
2-
Y/a
-n e- 29)
6, d 0 .
n
2
)
060.
m,
Jo '
r
3
0
nr
C20
+ -n r6e
/6
20r
4
16
+
+___
nreo2
':Z
7r 1 c
2,9[
co- 2
2 +y..e
77M01 9
2.
8
+
r 1-m 2cow z0
A'
_
n r 4co-sZ9
00
14
A 2.
xvft
do do= 0
*
A
iv.
A
We may now write down the second part of our third
approximation,
277(a
6?
nr3J&4
12..
4 fr 6'J
rrn r--n 9
,za-
c')
1c.'r
-+ n c'rJ--T9
rn zA- 6)
3za'
0a
7Cr4s6
+
2P
77rerM2 ,-a
77 r 4eosd 29
2q'la
Ao
2n
a.
(I
4
L
(a+c')
9
Ym
a+
2a.
neglecting all terms in
I
.
The integrals required to determine the value of
will now be worked out.
i.
.cnd~
d/9.
=
r
77
+.2
.
do=
-/
7Lr
7r
A
A
00
4
1 77,
S4n#
r zns
r
0y0
03
CO' (0
uTI
d
CO
a
o
n
_
(VJs1
_
6
_
0
vr rsxn 0
p)
0)
-
irmSn J9 +
4
771- M
n
nr 'Pn
I
)
C1
'r9
15
iii.
/dO
iv
a
d#
= 2 77/
3do
7n
~Z7)
nnd
A9
0
77M&
77/[/3f
L
9
g
z-
- #2]]
We may now write down the third part of our
third approximation.
=
list
em /n a.0
VqC
. I. C/'r 4
4
7r
c')4
4a
8
+
-nrr s n9
4r
s at
24a-
160"
A
77e'rn
_ lrrnut,
4
n e'rV + 77C'/n
-7n
J
4a
/60,
00-
which may be written
A
.
r
c')
3
rM
-r
4+
2
6a.
a
S3
h'''
..)
CL
Xs-V
2 a-
+ 3ec'
The integrals required to determine the value
of
now be worked out.
(.,will
- "&
/I.
/e'
A 0
r
-n
rn;e
d4 d'
0{
r
06
/0
277
2
m
/
co-s z ( 0- 9)
27
.e 2 #
od1b
t
~1
70
i-4
00
r
o
-
,
2
n
*7
-s 2(9-0)
o)
"f
,0
seT*
2.
d
d
G
e0o
c~o-s2('9)
Stq
' .0Ct4dck
ao a/ 0.
16
271
fr
,QI
(~ R~kr2~)
r
0
Id
£
0
77uvj 2!)
--
2
[(4
-n
)
+
+-ne e- I-i Z
I 91_
'c-os Z
32
36
36
+
77r
77 e 2m?
36
/W
dd
a
/
_ nr 6c2
lb
tn
/6
&
iii
rce 269
01
/+
1
co Z0
/6 .
h
ow
4{
i
We may now write down the fourth part of our
third approximation.
A-.
*w
t,1 a o<)
9-n ( a -ec')
(r.O)
-nI 6 * P
q6mCL
480a-
16
's4-O
32 ci-.
8-
o a o'
._
r 6A'
-nc'r
,t-
nk,2.M
/2
I
_ ne'sun 0
I
_r
terms in
288
S-,s4, 9
, m'
+
02
7r 6 oes 2O
256 a -
.8n1
/2 c
3.2(a~c')
neglecting all
..n r'
6
.
The integrals for the fifth part of the third
approximation are similar to those for the second part. We
may thus write.:
17
-,-
/
A 0 a cC
0) =
(Cr,
8n (a + C')
+ 77r 3 cm M
8a'
77 V 3M' XV4 0
enr c '*r
o2
_to a oc
+
4 29
m
IM
_r
Vx
41~
ja.
32 az
}
_m
SaQ
/16 a.'
/
3 z [a+e)
w em
t
77 r m en
11M
-n
Sa
1 aZ2a
SO,
.
en
8a
-7n
8a-
9
-n
,&
.2em
a
a.
tm 9
sa
neglecting all terms in-.
Combining the component parts for the third
approximation for current distribution, we have
.tv
. .i
.
(r.9)
^
' (r.19j
(r.
*
"
'(
'~
(re)
or
+
a
'(r,e)J=)
r
c'r
+
IK-
2a
23
-
+
9
_S_**
2a
.Sem-n
Km
-
,to a 'c
,V(a*e')
ia
-+
e'm'
01
24
~ICr
r7
.. A 0 a.ot"
-+ r
'<n9a
q(a~c')
I
+
a__
32 (a +c')
+
.+
*
0
2 a
.
3e'
'_
a.
r'st,
2 aa-t
+
A
L._
mr' 4
Ya
rr~-I'-
32 (a+c')
. o ae 0(
32 (a +e')
"Combining terms
tf
a
3
M's-
m-i
+
r 20
ad__
.
4a
9
m
Ya
7
18
ao
A'rj =
.Cie
+
{
o-
2a..
4/(a + c')
+
I rh2.
+
00a
r'
32(a+c')
.+
a *
+
a
e
I
+
+o
0 -
o a
/2a
4' a.
a
/
2-*
ao')
+ c'r
/2.
- 'mz
CL
+ 3to a o'
T_*
(a 4C')
*
Ci-
+
I-. ~C4~0
2 t- m 4S C<t 9 .?X
_rrm'''i~I 9
'
e4L
q 0i~a.
S
.
+
4:+n9
"i
2*a
32
rn
m
za
Aza
a2
A
+
a.J
Va
CL.
.
3 si~z 9
3 C'rn
.
,?a
+
2a,
l a,
2n
rI A&; 09
r
'a
7
c'Ir
{2
A
V
C '3?n
j
2a
a <
.
_r
17n
a
r 3 s<.A
+
'r2
+
3
n
_
fa
a
It may be preferable to have an expression for
The above equation-.can be
which does not contain C'.
transformed by substituting eS&c Of or c' .
o a
a
6)
-+ -
4
/
.
2 P M q~ " 6),,
.
+
a,
j
2a
4
r
Y C4
_
r m svu9
4 a.
/2a,
__na
Fa
4a
9 .
'9
__r_
2 a
Fa
a
6
+ -o a oc
3 2 6 a +r sih.9)
V M
_,.,
,
r m lSt
2-
9
z a,4
a
+
-+.
+.
r 3 S&'
y
t 3-
%'M
a
ra_
oa
19
Combining terms
kcr9~
a.+rsoc44
a
4(a+rct.9)
+s4
a
"4aciY
+
f
a c4
+
as
'+r
4'a
Saj
32 ( a ( O
3 2(aneo
+
+
_r____
3a-
2 a
fa
a.
In some cases, for instance if
the above expression is
to be integrated, it may be desirable to have an expression
forA
which does not have a
factor. We may do away
'
a - this fat
with this factor since
I
(
+ .r_
r "4
a.
We then have
e
+
M1
Za
.4
i.,Z rx-<A 9 +
a
S,4
_ _
ga
a
a
'a][
+
4
0(
+
'...
r 3
a
neglecting all terms in
f
___f
S"''-144
a
12*
a.
.j
2mr'-
99
**
a
la
-3 r 3[w 1 19 +
Ma
3
a
a
a
4 L
nl
0
t+
o0
Fa
a
+
+ a
a.
a
..
+ 't
a
a
rma rv-
'a
a
4449
+r m
3 a.
m
za
-
3,i
L
~~7j
C
i o
r
0___
+
0A
0
?aJ
_
b:
r M z 5+d $
f]
20
Let us consider the following terms of the above
expression
Undoubtedly,
if
the process of successive approximations was
taken a few steps further, additional terms of the above
series would be obtained.
e
These three terms are sufficient
to determine the nature of the series, however.
4
Z
oo
0o
l.+
(e
'
r'
This absolutely and uniformly convergent power
series is the development of the bessel function of the
first
kind and order zero for a variable ocer
M.=0
i.e.
4t
We know that the alternating current distribution
in a long straight wire of circular cross-section and of a
homogeneous; non-magnetic nature is given by
A 1
=
C/S7)
where
S
U
f
6~
But
Therefore
Ar
(
) = Ao T
(s
f).
21
Thus the expression that we have derived for the current
distribution in a ciretlai' loop of round wire can be
considered to be made up of two parts:
a. The bessel function which expresses the alternating
current distribution in a straight wire.
Some curvature correction terms.
b.
We can thus write
Ac.V=
+
-
o('QM
'~~'o
9
.'rjsV'
+
O
L9"
rm
+
e
_r'___
+
$
a
a.
t:
__
_
+
-
s
01
2a
a_
a
-
r
-rscre. 0
6a
32
e'a
O
3t' X"
**-
o
~rs3 M
-A"
Jal
a
Since the total current in the loop is
I
=f
t(,-)
r
d d '.
none of the above curvature correction terms will contribute
to the total current.
We have
T
=
.A.'
o
=
7
Jo (js ?jr) r ded
0
rd
o
/J~jY
S
where 3 is a bessel function of the first kind-.and first
order, defined by
z 1
If
z
2,z
I
2 !!
the value of I
_
-z4 2! 3!
be known,
_
_
_
2 6 3!1.
the value ofimay be
determined from (5) and substituted in (4) to find the
current distribution in the loop.
(j-
22
For the determinationof the watts loss and the
resistance ratio, the process is
as follows.
Let
Then by equation (10)
H
pw
of J.
=
R. Carson s paper
,
ep
-a
where A, is the permeability and for this case equal to /
HtrJis the tangential component of the magnetig force
due to current in the loop.
Then
A
=
/WA/(y.)
L
Ao
4L
+
y
-
2a
6a
3
+
(3
= Soa-gs
4
4;
i
2.
t
a
67
_T_
4r7[a
-
(
o
64
2I]
-5 3-<.c
9
9
-
&a
7a
___
+
aL~
a
J--EO
~of
32
+)n
o crrsoA
a
X/(Sr)-
0Z~ +_2_____rm_2
3r 3 J'n9
3.
of
i
4
o
a
a.
a
3
2
a
mi~
6 c
~fS44
2a
a
By equation (18)
a
of Carson s paper, the true
Wave Propagation over Parallel Wires:
Effect.
SalI
---
The Proximity
'
23
energy transferred to or from one centimeterof wire thru
its
resistance lose and is
equal to the
is
surface according to Poynting's theory,
P- 211
)
II\ = real part of -qf71
(-> 9)
9
(4-.19
o(.
which involves only values at the surface of the wire.The
term e(..)is the conjugate of e1 a.
6 ), taken at the surface.
Let us now determine the terms under the
integral sign.
H/ .. ) = -s
7Tl
J<
w
)
o '__
-
_
1wa-
32/w
2
6a
a
a
___a
l
Combining terms and transferring
from the
denominator to the numerater.
|-
-
T
Zo o~
0A)
f(j'Sqm)
T
r
+
W CL
-
2
q
+
U3 2 wL
3a
ii
2mn
The value of e(,, ) is
A
J
-A
+ 4o O~-o<
3M
(a S S
-
3
+
344t
_____
44a
7
32L
g$4IVtA
a.
)
-
Mo
a
3
ea
a
a
m
J
8a
24
Combining terms
c0d
-e
fw
UJO
q- vy_____
-L
a.
+
X,
_
6-0<
Sim
+
a0
3'L
30
JL
In order to determine the conjugate of this expression
we must know the conjugate of Jo(.S
rVI. We know that
Thus
J.
i
7~)=Tosm.c(I +-__
The conjugate of
is
This can be shown graphically. Considering a vector
of definite lenght as shown in Fig. (4),
swings it
axis,
from its
the operator irT
original position along the horizontal
in a counter-clockwise direction thru an angle of 1350.
The vector now lies in the second quadrant. The conjugate
of this vector occupies a similar position on the other
side of the horizontal axis. This conjugate vector lies
jrjV
F,7. *,
c/
25
in the third quadrant and makes an angle of 45*with the
horizontal axis. However, this vector can be obtained by
applying the operator-j
f6
to the original vector. Thus to f
find the conjugate of an expression which is multiplied by
provided
the operatorjFi one must change the operator tothis changes all the
j's
in the expression to-
We may now write down the value of)
Z~s~g
-:
,
O
Jo
)
(- sY6
lo0~mS.
-
6- -e) SIyis
The product
S
{
-
19~
9
N3
9 4
o(
+C
L
4 14 .
9
4
0.. 3
6~Sf
A
3J
(jS .?) 4
sf
3
6Soo~ AA
9
3O
3
~0~o ~
0
-
Z.o
Iot
d~ST
;
S
nnued on next page.
___
33
26
t.
4+a
C.
,
- oJf
A
+[-- 5o6~y
\
.
5-o
jX 0 6
VW . + __ M'")
-
+S
5
C
4-
+.,-m&_
C.
x
3x-
8a
J
2
3.
3O
8L
At~AI
32o u
32,
o
.
ss
4
[
CL
L J
This equation is of the form
A +3 -B
+C
where A, 3 ,and C are the constants in the above expression
for
,,
N
.,a.
We may now solve the integral
217
~-~+3
(A
clA9
+ CSAM')
0
A
-LOe
Tn
+0Sr2
2 A+ C).
Thus the value of
2.77 .t
0
2.77
d J
~
LI LT1(,s'h.JF
rn-F
-
'Hi
+
71m
;
wq
..
32to
0
3V
2.
a.
I o
(/*fl-;nJ]
-!!. A&
CrAa .
2-
0
ks-o
3 +- -i
+
ffa
W
m
31
4
-
L
.2 M 4
FC
.
27
It remains to determine the real part of this express&on.
From a consideration of the nature of the bessel functions
J0 (x)
,~ -x
2.
=
22.
and
Z
it
and
23 2!
2 '2!3!
will readily be seen that for both series
Q
J,
{-S' S,
mJ
the real part is the summation of the
( Sm)
the summation of t1e even
odd terms and the imaginary part is
terms.
The real part of. Ic
-ec~ 4.eJ )i(~~,eJ do
Lj 77
2W
--.panAQ Zo(+02_.
i*
Y
-+
SoC,0~*
Q-
Sa
*n.
a
-*
*
4W
-
5
Ta
Q.
r
(r-
a
w
M
a32W
at
r-e2
6/VV a
*I
ak
L
3a)
3
4
T-
.r
t
.
{ s
L
/6 a/i
2
..
o36a
t pa,
continued on next page.
Ja
28
-L~T
I
_ca
a
2.
-
~
-
Io Co<
/ a
L C,(6 M a
a
P..c6M14
i, r3 6ac2w
aoI u''
6/L
4tL
2
.
L -M
2-
r
3
v
5 2
-1ar
4- (
w
a
d'p'
a
25'6 a'7-WO
06
4tM
a
t-1/e4
do (-
m
to 1 -1
C0(4h M
z
a
r-1
2.
2 -6az2
aC
LD
00,
*-d
I ,r 4
(
)
0a,
3 0 72 a'U
'1to
)
Wherever " imag, part of " occurs, it is understood.
that the
excluded. The above expression gives the
is
resistance loss per om. of length of the loop. 'itmay readily
be put in a different form.
It is usual to put
/- (;s r)-
bus:
bers'r+
the conjugate of which is
7 (-
s
)
-
beisr
ber'r +
be-t.
be re
In general,
0
(j r;t-)
Then
/
e
LOds
osLjs
r) }
( ber'sr +'e'e
oi's
[(js )r)
-
29
and
Tr
j
=
et- 'sr +'/A'SrJ.
(
Keeping th&se transformations in mind,
it will
readily be seen that
I(
er sm){be
=
2 to
3
-4o +
Ao
0,A256a.
8a
3A-0z
A4
-4
-
**a-<'
wJ~36
32 a
-+
j
I
S
- ber s
1
32 aim
be 'S m
d
h r's
3w
la
256 a 1w
3o7 2 a,,-u>a
The loss per cm.
bei
length of the loop at zero
frequency is given by
The ratio of the a.e,
losses and d.c. losses as
determined by the above equations will give the skin effect
resistance ratio of the circular loop of wire under
consideration, However, it may be preferable to have a
formula giving the ratio 1!directly.
The total alternating current in the loop has
already been determined (p.21) by integrating the current
density over the cross- section of the wire.
be determined from formula (17)
nn .T =
It
will now
of Carson'a paper, i e.
H,..e
en
In our case, it may readily be seen that all the
30
terms in 9,V 0 will not contribute to the integral.
Thus
27
... fso>5-
T77
A 0 6T
crS
= .-
2
II
m
it
|
W
V m)I d6P
('s
s-~m)
identical to the expression for I previously obtained.
which is
This may be written in terms of ber'sand
- .-
I
( ber ',
.to~SLJ
beS,
i.e.
be0i'-m)
and
I
Ao~sm 1
( ber'.s .-
:i. w
I.,'
V
be-f..
Hence
IfZ
i.y
-s
((be
>'
's)
's)
. + (6e
Thus
o' 6'a. m
RI
q,
32 x
4
~c
S
-L
.
. 1
.s
h-e
4U
'
I(rIS
.
4(
2
S
2. C 6
lif44
;'I
)I
9'a
2d6 a iz'
I
I
L (bee
7 -A-'
's s")
{4(be
"S.m
J
.
.-J er2
be r XU 6 e*'S -&-"
(ber 'svm)
+
9'
+
L (ber')
.1106
3o72 0.w
L
q
W71'is
O1_
'
w.
b e,4 ' -< ) 2.
7
0C~ cOPU
a -s -s.1
A
- +(
b
(
+(
'.1
31
Putting
the skin effect resistance ratio
,
-na -
is
S erL
3 +
[4-
(ber's4j'- +(
IM
__
y"
of
.
e,
''"S
*
o4
So-
qg 6
J
Fa
I
t a'0 sMr
b
S4 .1
-n
s-)
(b~e
In Scientific Paper No.
,e.-sr-, bei'sb
2]
169,' the coefficIent of
be.
... be.r'si,
s--
+ be<;'s-4t)
( ber;'s-4)2
where
ben'-m eb
Le e s-m bei' s m.-
77 M L
-
11LO
In our case,
S
L70
FA-
Hence,
w\/
2n7
24 Ism
?hL.
de
er (
M
-no
b_
-
/7LO
6L
Since
s
=
4~v"?
and
=.
C4'.
/Vw
C-
?
our expression form may be simplified further.
Thus-i
TI'll
RI
-
3
be-
_-1 6 -n
s, n
bei 'Sm
( toe r')
wl1n
.er
:s
el<, be4 s
2 + ( b e' sW) I4
6to'
>M
_w-
k-1.
-n12-LolW w
46 a'--A...
a
.
?a 2-0a- 14 77 W
(b e'm
-f(bei
's
4J)
32
2'
4
(,
_
_
W aL
IT
lobe
'
-
m
+
T* w11m
b er S
w'
144
36-
ber '&")
6&
2.
*
<
X-b i
g
7/
2.y
0Ja
Since
b=2mg"'2 and
&'=
/
et-
9
the above expression may be further simplified.
b
'R
'R
L
2
be'b
44btr[kew'
bere'b)* + ( be:'6)b
- I M6
IL
32
1
-+
|a
(bei'
where the first term is the expression for the skin effect
resistance ratio of a straight wire and the remaining terms
are the curvature correction terms.
The quantity
br b be'b... be.r'b'berb
is tabulated in Scientific Paper No. 169 of the Bureau of
Standards by E. B. Rosa and F. W.Grover, page 226.
The values of ber'b and
'%are tabulated in a paper
"A Precise Method of Calculation of Skin Effect in Isolate d
Tubes" by H. B.
If
ber'b
Dwight,
Journal A. I. E. E. Aug. 1923, p. 827
and, be i'L cannot be obtained from the above
mentioned table, they may be obtained from the following
seriee
33
qX
4
bev'
ev 'X
=
3
7x
+
.
0
+
2%.
2% 4'
(-,2
2-*
X.
2
~t
F z4*
Tlese series are always convergent but the calculations
become laborious for large values of x.
Denoting(bev'x) +(bti&)
2
by
Y
,
the expansions develope#
by Russell and Savidge ( Scieti fie Paper No. 169, p. 176 )
give very accurate results for K? 6. These equations are:
2 (8X,)
2 (8X3+a-
NUMERICAL EXAMPIES.
( Slide rule calculations )
Ex. 1.
Let
:
I..
c-.:.
e-i4..
2.54
322
Z-41,
4
-
Then
.. c
~o
*c
(cooov,.
I,
= 2x - 4 1 4 x Zx 3.14
's Ioo o
11 24
169, p.
From Scientific Paper No.
h
S
6 ei - e
e,
6
1-(+ ( be ')
26
=..3'
226,
J
ooo 4 5~
The curvature correction is
3
-_.531 4
-'x2.54
I
;1',K4 . 531
6ev. 531
. ol94
8~ 8' 2 .64
x
32 x 1.641
The values of
A. I. E. E.
x
.199
. -531*L
.0194
&
.31
(bev'.-3-7)
6er' -S1 and
b t.
6' + (b ' E31)'
31 aa given in Journal
Aug. 1923, p. 827P are
= .. .0o A q \I
cAAd
6''
.5'31
Hence --
-
14o 1 4
3
(-. 246)
:Z. 0001
+-o1\9 =
.o'3 2
35
Substituting this value of
Y in
our curvature correction
- 3 - 11.3 x Io~% + 9.1 1 10~4.-5.54) - Io4 o xI O"(' -5.4.)
3
15.xo.
..
I
x
- .oo+ - 2. -
3. 3 2 bx
j
. 3\95 j
\12.0
4
\5.?-
Let
1.54 cA.,
I i*.::.
.=
=
=.o 1 94
eA.
,
4oo ooo ~.
Then
qK D~
3.14x 2 x 3.14 K too
1 p224
From Scientifi~c Paper No, 16.9,
000o
- S. 3 1
p. 226,-
ei' - bey'6 6 6
b e
( bev-'6)& +(.be.'
'6)
= 2,.-13
The curvature correction is
{
Ii
-
5.314
4?
+
Fx 2.q
)1A4x .131"
A ug.
E. E.
bev '5.31
-
.3'
8
L
x 2.5L4j
I)I
.01
3~ Zx 54
The values ofty'5.31
A. I.
.
.194
C.3'1
1923,
3.21')
aner'.
and
be.'
31.)
+ ( b el ' 5. i3 d
7.3' as given in Journal
p.827, are
o"&.
es'' S. 31 =- .(.53o
Hence
(.. - 3.2'11
L+
(-G
f
o
-:l.
3 + 4 2.:-=5
33
7
36
Substituting this value of
t
- 3
-
11-3 - 4
7..
3BZ,3 x 152o
5
3
3
.-
3
1 95
I
in our curvature correction
Y
IS 3o}
5'3 31o j
= 9 3.
1o
Let
3 z
a.
=
1. 6 i,
Z -I C
T
.44A.
AA.
I
16
~ =.
\1 1 X-
. tAA
@ 2o 0 C.
Then
Y. .
3.14 x 2 x 3.14 x I o o
g
o.014
1112 4
From Scientific Paper No. 169,- p. 226,
er '
-
b Ib
=
I.
o>010
'b6 - + ( bei' by e 0J
(b'e
The curvature correction is
+ i.oI4 4
- 3 .. .014
?1.61
1
t
x
\0
SThe values of
A.
I. E. E.
6ev' l.o1-:
1.014J
YA.62J
1578
?E
bCe- 'L
1 .0, 14
+ ( o -e.' I.0o 4)
ev'I.o'+andeL'I.o14 as given in Journal
I}
Aug. 1923, p. 827, are
-.
011
.4AI
.ei. 0 14 =
4 . 53362 .
Hence
(..0111i
+
(.6332j
Substituting this value of
-
.+ 2 5
in our curvature correction
37
{
+ .0 13 Fq (-5.95)
'3
.
0
-
3
16
5.5
(-
*o2.'l~
'L1 0 ~ j.
-
I
.314w
3 -. oZ18
-
-
3.(,ol K3-011I,
' o0
- 349
- . 9A l
.oSZ5
21
.11
...
B 4c
Ex.
4.
Let
a:
et.4,
'l.6
3A".
A.
Then
4
too 000 ~
I
From Scientific Paper No.
[b
169, p. 226,
6+C- t2 b Y
i.'
a
+.
L
(. bey'b,
The curvature correction is
{
I
)
Io;4
4
1i 2'4
x
o-19
= o-111
?~.Iq~2.)c3.I4xlc,
I'124
0 o..tyQ
9 t1 x
'.x- 5
1.b2
( b'er'
3 -2 y 1.( 2 '.
lo1
.1 + ( bel' lo.193 "
Y1'o-*14 and 6,&' 0-1 4
The values of
from Journal A. I, E. E.
f
cannot be obtained
Aug. 1923 p. 827,
We may use the series
'I,
-
x4
2-Ti
Thus
1 -
+,-t
YX
r.
+
('x)'s
...
3
11
38
9
2.x 3.14A
I .. .04 993 4-,
5. ,
-
.14-x_10
r-
cIt.14
(3
in our curvature correction
Substituting this value of
{I
73 - 2-1? (ob
5"vW
1-1v
oo
-15
. \4 9to
.3
5'(-3oD
9 SI Cj
1474x1-111
5x
00 \
~...
1 54'J
v.10'34P
1
..
ii
-10
L5~6 ~30oo
ANALYSIS OF EXAMPLES.
Undoubtedly; some of the results obtained in the
examples just worked out do not represent the actual facts.
These results must thus be analyzed in order to determine the
range of usefulifess of the skin effect resistance ratio
curvature correction formula which has been derived.
The derivation was only carried out to the third
term of the skin effect resistance ratio formula for a straight
wire, It is thus not warranted to consider the curvature
correction formula to be applicable in cases where the first
three terms of the straight wire equation form either a
or a series which is not sufficiently
divergent series
convergent.
for a straight wire which is
An equation for
very suitable for rapid calculation purposes is
given in
" Transmission Line Formulas " by H. B. Dwight, p. 114,
+...L-
1I2
x---X
19o
+
1ooto
where
10
-9
r1
-R
where
X
40
We will now calculate the values of - for a straight
wire corresponding to the curvature corrections obtained in
the four examples which have been worked out.
I.
3.it+-
o194&-
R =
2 x 3.t9 x, tooo
AA.
RI
x 1-124
1+2~.
+ -.
X
4
,l %.
12.
-
Oox, lo~ 9
4
ISo
IS
o-o-oe)
-
b
ooo43
.
.0.2
~
12
= 1. 0-4-
II.
= 121
il
' ++
1.21
-
L
IX
I
- 9.6.
lI.o
4.33-
III.
2 x 3.14
=
xlooo
.12+
x
lo'~ x3.\4 x .I588
x\o-
228+
I +
IV.
2 78x
I
'R
-I-
10ooo
Io000
____
12.
= I + G 9-2 - 3?2o
2F.?o
= -
31'~o,
19
-- .
.
0721
-01 2
JIco
i.3
41
Comparing these values of 1- with those previously
found., it is readily noticed that in the first and third
examples close checks are obtained while in the other two
examples, the differences are rather astounding, It would
thus be expected that the Ourva/,ture correction term
the first
for
and third examples are nearly correct while
those for the other two examples may be far from the actual
facts, Common sense alone will tell
that both q3. 0 and
,
.?1
are impossible results, the former being too large and the
latter too small.
t immediately appears that the curvature correction
formula which has been derived in this paper does not hold
for very large frequencies.
Whether or not the formula is
applicable must be determined in the manner shown in this
section.
CONCLUSIONS
The equation for the skin effect resistance ratio
of a circular loop of round; non- magnetic,--a-homogeneous wire
derived in this paper is
[
'.
-
lher' b6eL',
b__m'320a.
-
6)
-
4-89(
._-
.. er'b
±i
')-A
-
8
..
1
( b&kb)1+( bi
b
b
so-
1(a.)
where
a
radius of' wire in cm.
o~ radius of' loop in cm.
frquency in cycles
o- resistivity in abohms per cm.
The first term is the expression for the skin effect
resistance ratio of an isolated straight wire and the remaining
terms are the curvature correction terms. Although the total
expression for the skin effect resistance ratio of a straight
wire is here given, only the first three terms of the series
have been derived in this,work and. the curvature correction
terms given correspond to the first three terms of tb
atraight
wire formula. The curvature correction formula can thus not be
43
used when the first three terms of the straight wire equation
form either a divergent series or a series which is not a
sufficiently convergent. A suitable straight wire series
for calculation purposes is
2
"R
Igo
where
Wi-
W,10~
LO
0~
From the very nature of the series ( b )
,
the curvature
correction is only good for relatively low frequqncies. The
use of ( a ) implies the use of ( b ) as shown in the
numerical examples worked out in this paper in order to find
out whether or not ( a) applies to the particular case on hand.
From the results obtained in the numerical examples,
it
would appear that the curvature correction is much larger
than expected.
The writer is unable to detect anything in the
method of attack or the mathematical work which would to disprove
those results.
4,4
APPENDIX 1
SOME DEFINITE INTEGRALS ENCOUNTERED IN THIS PAPER.
p
U)
An p
dn ccp =Th _
'h
3
ad#=n r
sa~~~-
5
77k~n1
(3)
33
2
p'
$
st;f
4 -j
(s)
p
d-o
al
(p = 77r
1/
Po
nE
-nT
s-il@, dn, d$
JA
12
+ -y
Ze
m ei9
8
(c)/
/Ids
77 r
~
36
JI
+
cd
a,/
('7)
71rkn
-
ercod2
32
cos q
d#
2.
(9)
4
&
/)
(9)
42~
(o)
0&4
6
XP~b95
d#o
=0
stu $ do d zo
77
9
45
0)
(iz
A9
/oD d,
cZ Z577 im
(is)(
#211)
d
"
#/ = 7v
"< 9
0
{/
(11)
27
2
7~
0
#d
#
77...
eo Z
z
0
,'
r
.,
= r
16
o/*
36
APPEND IX 2
SOME USEFUL INDEFINITE INTEGRALS.
(i)
dv = ., v
f
(2)
(3)
.Ft
- /V
aos eolsb x
sou(-~ a
4lw.
x
swbx ox
(ct- - b)x
a (a -b)
-as
-
=
(
(5)
cv44.em
.
v
b14.
2 (ca + b)
(a-1J
Ss-a( a4b)x
2-(a
2 ( a - b)
bxdA=
..
"t, n/x
4w1,
Co nx
CSC
x d:
n#
<.(x
Lrh
C-$
I
14 ivx
7n+in
?
2.
cl
L
x
-e
!+
+
44
-x
X
-d]
C-Qs ( -, -- )aX
+
+ 1n-I
(g)
1,2.
+b)
4Xcx
+ 1't,
(?)
c~
1+*x
.~in
!+
0--0--
(()j
V
-
2 (a
()j
2 ( a.+b)
-b)x
+(a
'/ a~b~
_ e- La.+b)x
( -1. --
I
xm
1
oe-
x
X
I 1w
S
ox clol
d-o-
x
1
-ix
X
X
47
(9)
(Io) /
ax dx = x
x
-k" (
.,x
r)'
d" 0 =
V ,t ?6
x '1-0-
(
'K
'Vt
1s1-y
/
.A
x
.
(&
x)
rdx
APPENDIX 3.
POWER LOSS IN A CIRCULAR LOOP OF WIRE CARRING D. C.
We will now determine the power loss in a circular loop
of round, non- magneticg homogeneous wire carrying current
at zero frequency.
The resistance eA of a circular filament having a trace
rce cr at M on the equipotential
surface under consideration
( Fig. I ) is
-21 (1 +
Scedet
The drop at
0
must necessarily be equal to the drop at M
Thue
7.-na'-
= 2,n ( 0. + YN
&k%01
) T- Z (Y.9)
and the current distribution in the loop may be expressed by
a,
The watts loss in the filament having a trace r dir oL 0 at
2
a
o-+ loop i
and the total power lose in the loop is
d
din(
r
YcdEccr
49
-Ftjd..4
(r.9) ct1.
2.71-
As bef'ore
Thus~
Y-
=
2T,
o
a.
a..
o
(
(2. Sr r 0 . io'
&
02.n )
l.. ..
2
It is usually assumed that the current distribution
is uniform over the entire equipotential surface and the
resistance of the f&lament passing thru
D is
taken as the average
resistance of the wire. Under these conditions
which is
If
identical to
more terms of the series
0.
were considered in the determination of
that we would have
,
it
is
evident
50
Since the series was not carried any further elsewhere
in this paper, it does not seem that it should be done here.
However, as a matter of interest, we will determine the value
of
, ' for the following terms
a%-
OL
Then
~P,
a2-n o-
9
CL
+
i
.
.,
2C
r
3
of7,)oycowl
qCO?
4 x'3-ni
'P2
The
L
r s
4
rThe
L
6,
+
=+l-2
Y.
-Pif
3
.
lsote
.o- t
oFnow
The increase in power loss due to the uneven current
distribution in the loop is thus relatively small and may
be neglected for most purposes,
Ot9
APPENDIX 4.
POWER LOSS IN A HOLLOW DISC CARRYING DIRECT CURRENT.
We will now determine the power loss in a strap of
non-magnetici homogeneous material of the shape shown in
ig.
5.
."et
inner radius
san.
outer radius
a= mean radius
0
thickness of material
Fi.5.
The resistance clR of an element of the strap is
It can readily be shown the current distribution in
the strap is
where
A= current density at radius o.
The total current flowing thru a filamenttof crasssection -cLris
a
=
0A
,Ycr
52
The total power loss in the strap is
thus
rc~
VA
a
is usually assumed that the current distribution
It
is uniform over the entire strap and the resistance of the
filament at radius a
is
+ML
'
taken as the average resis-
tance of the strap.Under these conditions,
=
Th
cA-
Thus
( w- -
)
_
1
-P,
9., Tcr o -
I. tV
'L
M2
,-.
EXAMPLET.
(a
Let
,=
M
3i
I
k"
=
4i
Then
A.,
PL
(b)
2.
-
~(1.7q/3-1.3I63)=
0 1 3
z
Let
SI
Then
?I--L
'~P
I.5'c.
6931=
1.0,3'7.
8
53
The increase in power loss due to
nneensurrent
distribution in the strap must be considered in some cases
since it
may amount to several %
of the total power lose.
BIBLIOGRAPHY.
" Formulas and Tables for the Calculation of Mutual
1.
and Self Inductance " by E. B. Rosa and F.W. Grover-Scientific Paper of the Bureau of Standards No. 169.
2.
An Integral Equation for Skin Effect in Parallel
"
Conductors " by Charles Manneback - Publication No. 30
of the Blectrical Engineering Research Division at
M.I. T.
and Journal of Mathematics and Physics,
April, 1922,
3.
"A Short table of Integrals " by B. 0. Peirce.
Advanced Calculus" by F.S.
4.
5.
"
Proximity Effect in Wires and Thin Tubes " by
H. B.
6.
Dwight- Trans. A.I. E. E.
"
April, 1921,
page 607,
Proximity Effect in a Seven Strand Cable " by
J. E. L. Tweedale, M. I. T.
8.
1923; Page 850
" Wave Propagation over Parallel Wires " by J.R.Carson
Philosophical Mag.,
7.
Woods.
Thesis, '-ourse VL, 1927.
" A Precise Method of Calculation Of Skin Effect in
Isolated Tubes. " by H. B. Dwight- Journal A.
I.
E.
E.
August 1923, page 827.
9.
" Transmission Line Formulas " by H. B. Dwight , p.114.