formulae, tables, and curves for computing the mutual inductance of two coaxial circles ·...
TRANSCRIPT
FORMULAS. TABLES. AND CURVES FOR COMPUTINGTHE MUTUAL INDUCTANCE OF TWO COAXIALCIRCLES
By Harvey L. Curtis and C. Matilda Sparks
ABSTRACT ^
The formulas for the mutual inductance of coaxial circles can be expressed either
in terms of the radii of the circles and the distance between them, or in terms of a
parameter which is a function of these dimensions. Both types of formulas are de-
veloped in this paper, and tables prepared to aid in the computation by each type.
Charts are included from which an approximate value of the mutual inductance can
be read directly. A number of examples are given to show the methods of computa-
tion by the different formulas, tables, and charts.
CONTENTSPage
I. Introduction 542
II. The nomenclature 543
III. Derivation of series formulas in terms of k^ and k^ 545
IV. Formulas involving only the radii of the coils and the distance between
them 549V. Use of tables to compute an inductance 552
VI. Determination of coaxial circles to have a desired mutual inductance .... 553VII. Use of charts 554
VIII. Mutual inductance of coaxial coils 556IX. Examples of use of charts, tables, and formulas 557X. Summary 561
Appendix—Tables and charts for determining the mutual inductance of coaxial
circles 562
Table i. Values of H in the equation M=r-^ H 562
Table 2. Values of H in the equation M=ri H when the circles are close to-
gether (P>o.85) 564Table 3. Values of H in the equation M=r^ H when the circles are far apart
(^'<o.i) 565
Table 4. Values ol k^ when D is less than A—circles close together 566
Table 5. Values of fe ^ when D is greater than A—circles far apart 568
Table 6. Values of P in the equation ^=~a P when D is less than A—circles
near together 569
Table 7. Values of P in the equation -^=-4 ^ when D is greater than A and
less than three times A 572
541
542 Scientific Papers of the Bureau of Standards [Voi. ig
Table 8. Values of S in the equation M== ^3 5 when D is greater than 3 A— Page.
circles far apart 574
Table 9. Values of 24n/|^N4 57^
Fig. I. Diagrammatic cross section of two coaxial circles showing dimensions .
.
544Fig. 2 . Chart showing relative size of circles and distance between them to give a
definite value of ~- General survey of a large region Facing 554
Fig. 3 . Chart showing relative size of circles and distance between them to give a
Mdefinite value of -r-' Important region in detail Facing 554
M D aFig. 4. Chart showing the relation ^f -j- ^<* T for different values of -r ... Facing 554
I. INTRODUCTION
An exact formula for the mutual inductance of coaxial circles
was derived by Maxwell ^ in terms of elHptic integrals. How-ever, it does not readily lend itself to computation either whenthe coils are close together or when they are far apart. ^ For this
reason a large number of series formulas ^ have been developed.
Some of these give accurate values of the mutual inductance whenthe two coils are close together, others when the coils are far apart,
and some in intermediate ranges. For any given case it is pos-
sible to select one or more formulas which will give an accurate
value of the mutual inductance. However, most of these are
somewhat compHcated, and difficulty is often experienced in
applying them to numerical examples.
By using several different formulas to make the necessary com-
putations, the Bureau of Standards has published a table * which
provides a simple method for computing the mutual inductance
of any two coaxial circles. The formula for use with this table is
of the form M = -y^Aa C, the values of C as a function of k' (defined
later) being given in the table.
1 Electricity and Magnetism, 2, art. 701.
2 Computation by this formula has been simplified, and its range of usefulness greatly extended by the
tables of elliptic integrals recently published by Nagaoka and Sakurai. These tables, entitled "Tables
of Theta Functions, Elliptic Integrals K and E, and Associated Coefficients in the Numerical Calculation
of Elliptic Functions," were published by the Tokyo Institute of Physical and Chemical Research in
December, 1922.
' These formulas have been collected and their limitations pointed out by Rosa and Grover. B. S.
Bull., 8, pp. 6-32, B. S. Sci. Paper No. 169; and 14, pp. 538-544, B. S. Sci. Paper No. 320.
< This is published in B. S. Circular No 74, entitled " Radio Instruments and Measurements," p. 275.
The table was prepared by Dr. F. W. Grover, one of the authors of the circular. However, the methodof computing the table is not indicated.
spaX] Mutual Inductance of Coaxial Circles 543
In the present paper formulas are derived and tables of con-
stants given for computing by two methods the mutual induc-
tance of any two coaxial circles. The tables can also be used in
solving the inverse problem, viz, determining the dimensions of
circles which will give a desired mutual inductance. In addition,
charts are given from which an approximate value of the mutual
inductance can be read directly, and from which the dimensions
of circles to give a desired mutual inductance can be determined
by inspection.
II. THE NOMENCLATURE
The constants which are generally measured, and from which
the others are derived, are A, the radius of the larger circle; a,
the radius of the smaller circle; and D, the distance between the
centers of the circles. From these are determined the constants
used in computation, viz
:
r\ = (A+ay + D' (i)
r\ = {A-ay + D' (2)
4a
k2^r\-r\ 4Aa A
r\ {A-Vay-\-D' ( aV J^ ^^^
y'^Aj^A'
( _^Y ^^ ^ ^ "r'^'iA+ay + D"'/ aV D^ ^4;
h4gA /^aA
'~{r,+r,y~[^(^A^ay + D' + ^{A-ay + D'Y ^^^
544 Scientific Papers of the Bureau of Standards wu. to
-Axis
Fig. I.
—
Diagrammatic cross section of two coaxial circles showing dimensions
sTa^] Mutual Inductance of Coaxial Circles 545
WhenD = o
When Z)?^o
When D is large relative to A
The above is the nomenclature used by Rosa and Grover in
Scientific Paper No. 169, with the exception that D is used for d.
The geometric meaning of r^ and r^ is shown in Figure i
.
Since ^' =— > and since rg is always less than r^, then k^ is always
less than imity, which value it approaches as a limiting case. Like-
wise, all values of k lie between zero and unity.
The other terms used are the complete elliptic integrals^ Kand E. They will be defined where used in terms of series.
The natural logarithm will be indicated by the symbol "logn"
while the symbol "log" will be used for common logarithms
only.
III. DERIVATION OF SERIES FORMULAS IN TERMS OFk' AND k'^
There are well-known series for expanding the elliptic integrals
However, the application of these series to the elliptic integral
formula for mutual inductance is often a tedious process. A simple
transformation will simplify the appHcation.
The elliptic integral formula as given by Maxwell is
M = 4W^a[(|-^);<:-j£] (8)
6 The complete elliptic integral of the first kind is represented by K instead of F, which was used in
B. S. Sci. Paper No. 169.
546 Scientific Papers of the Bureau of Standards {Vol. ig
From equation (3) 2^jAa = k r^. Putting this value in equa-
tion (8)
M-=2Trr^{{2-¥) K-2E] (9)
If the expansion is to be in terms of k'^ (equal to i — k^) then
M = 27rri[(i+^'2) K-2E]
The series values of K and E in terms of k are ®
x= IT
'2
00 {2ny24n
(^^Yk'^
IT00
. (1:2ny
IE= — .^^2 n=o 2^"^ (1^)* 2n — l
Substituting these values in (9)
)^2n
(10)
(II)
(12)
(\2ny 2 (2n + iy
8 ^n^o2^''(\ny (n+i) (n + 2)(13)
This^ may be written as M = rj_ H where i7 is a function of ^^.
When the coils are far apart, or when one coil is much larger than
the other, k^ is small and the series is rapidly convergent. In
this region H is very nearly proportional to k^.
The expressions for K and E in terms of ^', given as equation
(3) of Scientific Paper No. 169, can be written in the form
i^ = (logn|,)[i+i-U--fi:.|^- +
.1' 3^ (2n — iy , ,,
{2ny
[2^1X2^ ^2' 4^ViX2^3X4/
{2n — iyC 2^2,^ 2 \i" {2n — i)'' / 2 2
"^P {2ny ViX2"^3X4
= s{\2nyk'^^ ^- 16 {2n-\-iyk- 2^Wny i2^''^V' ^
•]
4 {2n — i)2n^
(14)
« B. S. Sci. Paper No. 169, p. 9.
'' This equation (13) is the same as equation (5) of B. S. Sci. Paper No. 169.
spl^s] Mutual Inductance of Coaxial Circles 547
^2' 4= (2TO-2)' 2W ** ^ ]
+ ' 2ViX27^ 2^ 4ViX2^3X4>'
l\3\ (2tl-3)^(2n-l) r 2^
^ .--
2^*4'*"(2n-2)^ 2n L1X2 3X4
(2n— 3) (2w— 2) \2n — I ) 2n
J
=s (|2n)^Aj'^Y ^J^ r n . 16 i^
n)* [2^-1 ^^jb'^^(2t^-;?^ 2*^ (|w)* \j2n—i ^ k'^ {2n—iy
(15)
° 2(W+I) ^(^+1) (2^+1)
J
Substituting these values in equation (10)
I i\3\.. A2n-3r^2' 42 6^ (2n)2 '^ ^
— 2^24 ^2M\iX2 2/'^ ^2^2 6\iX2^3X4 ef
^'"^2242 6^ (2i^)^ ViX2'^3X4
^ ^ ^_^W+.. 11(27^ — 3) (2^—2) 2il/
JJ+
r2D.
TT^S00 Q2nyk
n2k
^r__j__. 16 4_
_ _^ 1(n+i) {2n+i) 2{n+iyl^o {s+i) (2^+1)
J
102423°—24-
548 Scientific Papers of the Bureau of Standards [Vol. IQ
where H is a function^ oi k'^ and hence of fe^ This formula can
be used for values of ^'^ from o to 0.7, provided a large number
of terms are used. When k'^ is small logn p^ becomes large, and
the summation depends almost entirely on the value of this loga-
rithm. Hence, in this region H is nearly proportional to colog k''^.
The mutual inductance of any two coaxial circles can be com-
puted either by formula (13) or (16). If k"^ is less than 0.4
{W^ greater than 0.6) equation (13) should be used. For larger
values of k^ use equation (16).
By writing the coefficients of the powers of k and k' as deci-
mals, the above equations are in much more convenient form for
computation. Equation (13) becomes
= 1.23370 k^ 1.00000
+ .75000 ^^
+ .58594^'
+ .47852^'
+ .40375 ^«
+ .34895^''
+ .30715^''
+ .27424^^* (17)
+ .24767 ^^«
+ .22578 ^^«
+ .20744 F«
+
8 Formula (i6) is equivalent to Butterworth's formula D given in Phil. Mag. 31, p. 276; 1916.
as formula (4A) in Grover's collection, B. S. Bull. 14, p. 540. B. S. Sd. Paper No. 320.
It is given
Curtis 1SparksJ
Mutual Inductance of Coaxial Circles 549
Writing all numerics as decimals, equation (i6) becomes
M = [8.71035 + 7-23378 colog k'""] 1.00000
+ .25000 k'^
+ .01562 k'^
+ .00391 k'^
+ .00153 k'^
+ .00075 k'^""
+ .00042 k'^^
+ .00026 k'^"^
+ .0001 7 k'^^
+ .00012 k'^^
+ .00008 k'^""
+ .00006 k'^^
+ .00005 ^'^^
+ .00004 k'^^
+ .00003 k'^^
+ .00002 k'^^
+
4.00000
- .50000 k'^
+ .01562 k'^
+ .00651 fe'«
+ .00300 k'^
+ .00160 ib''"
+ .00094 k^^^
+ .00060 ^'^*
+ .00040 k'^^
+ .00027 k"^ (18)
+ .00020 k'^^
+ .00015 k'^^
+ .0001 1 k'^^
+ .00009 k'^^
+ .00007 k'^^
+ .00006 k^^^
+
It is only in the middle range of values of k and k' that the
large number of terms given above will be required, and then
only when extreme accuracy is desired.
The principal use of these formulas is to compute the constants
of Table i . This table permits the computation of an inductance
with an accuracy of one-tenth per cent, and its use is muchsimpler than that of either of the above formulas. It is only
when higher precision is required that it is necessary to resort to
either formula (17) or (18).
IV. FORMULAS INVOLVING ONLY THE RADII OF THE COILSAND THE DISTANCE BETWEEN THEM
Formulas can be derived which give the mutual inductance as
an algebraic function of the measured dimensions. Such formulas
are suitable for computation in all cases, except when the coils
are close together. These formulas, while in themselves quite
complicated, permit the computation of tables from which the
mutual inductance can easily be determined.
550 Scientific Papers of the Bureau of Standards [Vol, 19
o\
CO
i
^
ICO
cjK
eK
CI.
oo
o
<L)
dII
o
I
t
Phd ^
+
Rl
Curtis 1Sparks} Mutual Inductance of Coaxial Circles 551
By putting all the numerics in the form of decimals, this equation
can be written
:
M = 1.00000
+ .37500 aVA'+ .23438 aVA*
+ .17090 a7^^
+ .13458 a«M«+ .11103 a'VA'^'
+ .09451 a^VA"
+ .08227 a"/A" (20)
+ .07284 a'7A'«
+ .06535 a^7A^«
+ .05926 a^/A^
+
The numerics of this equation apply also to equation (19) and
can advantageously be used when it is necessary to compute bythat formula. Equation (19) is convergent for all values of D,
except when a =A and D==o. It converges very rapidly when
-J is large relative to ^* In general, it may be said that by means
of this formula the mutual inductance between any two coaxial
circles can be computed. However, when the circles are close
together, the convergence is so slow that the computation is very
laborious. When the shortest distance between the circumferences
is greater than one-third of the radius of the larger circle, an
accm-acy greater than i part in 10,000 will be obtained if the first
II terms of the series are used. When the distance between the
centers is greater than the radius of the larger circle, the sameaccuracy can be obtained by using the first 3 terms of the series.
Since equation (20) is convergent when a<A and whena =A if Dt^Oj it can be used in mathematical derivations where
it is necessary to have a formula for the mutual inductance which
is applicable to a wide range of circles. For example, it should
be possible by means of this formula to compute the skin effect
in a circular ring by the integral-equation method. Whether the
resulting formula will be sufficiently convergent for practical use
can not be predicted.
552 Scientific Papers of the Bureau of Standards [Voi. 19
If D is large relative to A, then by substituting the value of k^
given as equation (7) above in Grover's series formula
4^£)4n
D3 (21)
Equation (21) is the same as equation (19), but is written in a
form which simplifies the construction of tables when the coils
are far apart. When D approaches infinity, the value of Papproaches zero asymptotically, making interpolation impossible,
whereas the value of 5 approaches unity in a nearly linear manner.
Hence, to facilitate computation when the coils are far apart,
tables of S are desirable. However, equation (21) should not beused for the direct computation of an inductance as (19) is equiv-
alent and simpler.
V. USE OF TABLES TO COMPUTE AN INDUCTANCE
Two different types of tables are given at the end of this paper,
each one of which permits the calculation of the mutual inductance
of any two coaxial circles. In the first type (Tables 1,2, and 3)
there is a single parameter which must be determined from the
measured dimensions. In the second type (Tables 6, 7, and 8) there
are two parameters which come directly from the measured dimen-
sions. In both types of tables the parameters have been chosen
so that the interpolation is approximately linear, thus reducing
the work involved in making a computation and increasing the
accuracy of the result. In the first type of table the parameter
is a function oikoxh' . The size of the tables for a given accuracy
can be decidedly reduced by choosing different parameters for
different ranges of h. For this reason three tables have been
constructed.
MIn Table i, values of — are given as a function of h^ (or h'^)
throughout the entire range of values. However, for the ranges
k^ = o to ^^ = 0.1 and ^^ = 0.85 to ^^=1, the values of — are
changing so 'rapidly that interpolation is not accurate. For the
region ^^=0.85 to ^^ = 1.0 (^'^ = 0.15 to h''^ = 6)\ that is, when the
coils are close together. Table 2 will give the necessary accuracy.
If ¥ is less than o.i; that is, if the coils are far apart. Table 3
should be used.
sparks]Mutual Ifiductance of Coaxial Circles 553
To compute the inductance of any two coaxial circles by these
tables, the following procedure should be followed: From the
measured dimensions compute r^ and k^. Values of k^ in terms of
the ratios of dimensions are given in Tables 4 and 5. If the value
of k^ lies between o.i and 0.85, find the values of H from Table i.
If k^ is greater than 0.85, then find the cologarithm of k'^ and use
this as the parameter in Table 2 to obtain H. However, if k^ is
less than o.i, find k^ and determine the value of H from Table 3.
It is generally simpler to compute the inductance directly from
the dimensions by means of Tables 6, 7, and 8. This avoids the
necessity of computing the auxiliary constants r^ and ^^ Table 6
gives the values when the distance between the circles is less than
the radius of the larger circle, Table 7 when the distance between
the circles is greater than the radius of the larger circle but less than
three times the radius, and Table 8 when the distance between the
circles is more than 2 . 2 times the radius of the larger circle. There
is a region where Tables 7 and 8 overlap. In this region computa-
tion is simpler by Table 7, but Table 8 gives the greater accuracy.
If the mutual inductance is sought for circles of which the values
of the parameters are not those given in the tables, a double inter-
polation will be necessary. Since the interpolation is approxi-
mately linear in both directions, it is only necessary to add to the
tabtdar value the sum of the horizontal and vertical interpolation.
In general, the tables are constructed so that an accuracy of one-
tenth per cent can be obtained in the most unfavorable case of
interpolation. Where that was not feasible, the tabular differences
have been omitted.
VI. DETERMINATION OF COAXIAL CIRCLES TO HAVE ADESIRED MUTUAL INDUCTANCE
In designing apparatus it is often necessary to determine the
diameter and distance apart of two circles to have a predetermined
inductance. This can readily be done by either set of tables or
by the charts.
In using the first set of tables, a value of A must first be chosen.
MIf it is desired to have the coils close together, the value of —^ must
be large, while if they are to be far apart the value must be small.
MOrdinarily a value of —j between 4.0 and o. i should be chosen.
Then since one value of r^ is 2A, one value of H is found bydividing M hy 2A. To avoid interpolation, select in the proper
Then ^i = -7r* By equation (3)
554 Scientific Papers of the Bureau of Standards [VoI.jq
Mtable a value of H, which is near the value of —j> and read the
corresponding value of k^.
1 (3)
k'r\a =—j^
4Aand by equation (i)
D = -^r\-{A + ay
If the above does not give a suitable size and location for the
second circle, other values can readily be determined. Suppose
that it is desired to have the second circle larger than that given
above. Select a value of H which is 10 or 20 per cent smaller
than the one which was first used and compute the values of a
and D. Should the value of D be imaginary, then too small a
value of H has been chosen.
If it is desired to have the second circle of smaller radius than
that given by the first computation, then the value of H must be
chosen which is larger than the one which was first used. Theprocedure is then the same as that outlined above. A few trials
will give the desired size and location of the second circle.
If the relative size of the coils and distance between them are
given, then Tables 6, 7, and 8 can be used advantageously to de-
termine the absolute size to give a desired mutual inductance.
If -r is less than three, the value of P can be found from Table
6 or Table 7. Then a == -75- • — » and the values of A and D can' P a
be computed from the ratios initially given. If -r is greater than
three, the value of S can be found from Table 8. Then
M Al D^^^ S ' a'' A^
In many cases this method will be simpler than that by the use
of Table i.
VII. USE OF CHARTS
Three charts (figs. 2, 3, and 4) have been prepared which are
useful when approximate solutions of problems connected with
the mutual inductance of coaxial circles are sufficient. As three
\
^— -—-^.^
"1 ^ ^-->
/'^ \
/ \/ \
/ \/ \
. / \
3.2
\ / \\ \
3.0
\•
/'^
\ /\
/ \ /
\o give definite values of M/A. General survey of a large region.
\"\
/M= Mutual inductance, millimicrohenrys
\A= Radius of large circle, centimeters \ /
\a lus sma circ.e, cen ime ers
/
\. /
\^ /
2.2
/
\, /\ /-
^ \ /
\\
/ \ 1 /
1.6
\ / \ /\
\ ^ / \ / / /\ \ \ / /
/ /\ \ \ / / /
1.4 N \ \ / / /\ \ \ ,^- / /
/ /1.2
\ \ \\
/ iyA=30 \ M/a=20 'rA=io / ^=5 / %'3 / < M/a=2
\ \ \\
/ ^ \~// / /\N \ \ \ \ (
/ / / MA=I/
\ \^ \ \ \ \ \ / /
/ / // -^
\ \ \ \ \ V y / / / y ^^--
~"\ \ \ \ N yv ^y y
-'y
r^"-.^ \^ \ ^ ~ ^ ^ / ^ ^ cr-S^
::-. ^^ ^^ ^^\ \ ^ -^^ <=-
-^ ^ ^^ ..-tTI^ --_^ -- "-- ~- ZI] -—
'
" ^^ -^t^'/.=2- *=-
~-~^ .__^ --. ^
'^— -— -^ -—— ^ _^^^ _——r—-__
~-^.^
-- -—- —— ___ --. ^ .
_- --jjrr' \— —- zz;-^ — — —————— -—
1
.
—
—
•
1——-— — — ——"24 2 2 2J} 1 B I6 1 4 I2 1 . 3 .(5 f4-
»5 , •^\ k .£ 1 12 1/4 1. s> 1.8 2.0 2.2 2.4
D/A
Fig. j
MUTUAL INDUCTANCE OF TWO COAXIAL CIRCLES
24 (Face p. 554, No. 2.)
— ————————— — — — — ——i
—V^
"~
n~r
MUTUAL INDUCTANCE OF TWO COAXIAL CIRCLES
.._
Chart showing the relation „f M/A to D/A for diflerent values _
°'"'*M=Mutual inductance, raiUimicrohenrys -
1A= Radius of large circle, centimeters
<3
^
\ r--a^=l.'o
\\\ 1 «^=oaV \
^
1 \M\L \
/ \^ \
/
/ %=o.a
1 j / \>
1 / \^^
/ 1 / \\
/
,
/ \ A// / \
\^
/ 1 \ \ %.1
/ \ \ "Tt/ / /
7^\ \ \ yA=a6
/. / / \ \) (\ Ui.i/ / / .^^ \A \
'—
7'
/y / y \ / \ V \^'^'\c^=0.4
//J / / /
/ \ /\ \ t s^ 2^=0.3
/V /I / y -__ Xs y/\ N?\ aA-y/ / / /^ *
< N ''
^v 4.^ y / U-_ \, ^
y^.\ -^%^^ ::d ^ ^ '-' •^ -^ -^
\
^^ >z 7 c^^^m ~ ~~~
a==s^S2 —::^ ;;:^^^ ^^ ^ —— u_
r^^~-- -^^ ;::
:--. Z::; =— —EmmctefflS^^5 ^=1 ;F=t=1
r=sE :== — 7^ M—J:—
,
-^ — -^=:== == ^^ ^samb I /i /. ^j •« iJ- ,2 (
Ii if > .£ 12 M- \&- r to 22 ^.4
>Al:f--l\ (Fm,> p.
sTa^s]Mutual Inductance of Coaxial Circles 555
coordinates are necessary to completely represent this function,
it is only possible on plane coordinate paper to plot contour
lines which show the curves of two of the variables when the
third has a definite numerical value. In the first two charts
(figs. 2 and 3), definite values of -j- are chosen, and curves given
for-J
as ordinate, and -j as abscissa. The only difference be-
tween these charts is that the second (fig. 3) covers a smaller
region with a larger scale than does the first. The third chart
(fig. 4) gives the curves of -j as ordinate and ~r as abscissa for
definite values of -^'
To determine the mutual inductance of given coils, the third
chart (fig. 4) should be used. Should the value of -j be that for
Mwhich a curve is plotted, then the value of -^ for the given value
Dof -J can be read quite accurately. In case it is necessary to
interpolate between the curves, the accuracy will be considerably
reduced.
To determine the size of circles and distance between them to
give a desired mutual inductance, the first and second charts
(figs. 2 and 3) are useful. In these charts the curves correspond
to the lines of magnetic force ^^ around a circle, the center of
which coincides with the origin, the plane of which is perpen-
dicular to the X axis, and the radius of which is unity. Since
the mutual inductance depends on the number of lines of magnetic
force from the first circle which cuts the second, the curves showthe different sizes and positions that the second circle can have
to give a single value of the mutual inductance. Another wayof visualizing this is to consider that the diagram is so reduced
in scale that the distance ^ = i is i cm. Then if one circle has
its center at the origin, its plane perpendicular to the axis, andits radius equal to i cm; and if the second circle, having its
center on the X axis, has the point where its circumference cuts
the paper coincident with one of the curves, then the ordinate
and abscissa of this point give the radius of the circle and the
10 Such a set of curves without any numerical values is given as Fig. i8 in Maxwell's Electricity andMagnetism, 2.
102423°—24 3
556 Scientific Papers of the Bureau of Standards ivo?. 19
distance between the circles, respectively, while the value of
mutual inductance in millimicrohenrys is given by the value of
-r on the curve.ATo use this chart for determining the size and distance of two
circles to give a desired mutual inductance, it is first necessary to
choose the radius A of one of the circles which will be used. It is
Mdesirable to choose this value of A so that -r- is one of the values
for which a curve is given in the charts (figs. 2 or 3.) Then by
selecting various points on the curve, a series of values of ~j and
the corresponding values -j can be obtained, any pair of which
will give the desired mutual inductance. If an additional condi-
tion is imposed, such, for example, as requiring that the circles
shall have the same radii (A = a) , or that the circles shall lie in
the same plane (D = o), then there are only one pair of values for
D . a
A ^^^ AIf it is desired to design an inductance in which the radii of
the two circles shall have a definite ratio, then the third chart
(fig. 4) can be used advantageously. For example, if the circles
are to be of equal size, -j = i, then from the curve so labeled a
series of values for -r- and the corresponding values of ~j can be
determined. This chart is especially useful where the relative
dimensions are known. Then if ~j is one of the values for which
Ma curve is plotted, the value of -r- is read directly from the chart.
VIII. MUTUAL INDUCTANCE OF COAXIAL COILS
All the formulas for the mutual inductance of coaxial coils are
based on the nautual inductance of two coaxial circles, the radii
of which are the mean radii of the coils, and the centers of which
coincide with the centers of the coils. If the breadth and depth
of the coils' are small relative to their mean radii and to the dis-
tance between them, then
M = n^n2Mo
spaJks] Mutual Inductance of Coaxial Circles 557
where M is the mutual inductance between the coils, Mo the
mutual inductance between two coaxial circles at the center of
the coils, and n^ and ^2 the number of turns in the two coils,
respectively. In this case, all the methods of computation and
design which have been given under coaxial circles can be adapted
to use with coils by a proper use of the factor n^ nj.
If the coils are close together, or if their radii are small, correc-
tion terms need to be applied " to the above formula. No sim-
ple method of applying these corrections has yet been developed.
K. EXAMPLES OF USE OF CHARTS, TABLES, ANDFORMULAS
To illustrate the use of the tables, charts, and formulas, exam-
ples will be selected from those given by Rosa and Grover,^^
who have given the results correct to about i part in 1,000,000.
For each example, there will, in general, be five solutions by the
methods given in this paper, viz, one by the charts, two by tables,
and two by formulas. However, the charts can not be used
when the circumferences are close together, or when the distance
between the coils is great. The second set of tables (Tables 6, 7,
and 8) will not give values when the circumferences are close
together, and under the same condition the formula—equation
(18)—giving the mutual inductance in terms of the measured
dimensions is not sufficiently convergent to be practical. Hence,
when the circumferences are close together, only two values can
be obtained, and when they are far apart only four values.
EXAMPLE 1
Measured dimensions:
A = a = 25 cm D = 20 cm
Computed constants:
r^ = 2,900 ^1=53.8516
r\ = 400 r2 = 20.000
)b' = 0.862069 k''= 0.137931
a
s=-Rosa and Grover value of mutual inductance is 167.0856 milli-
microhenrys.
" See Rosa and Grover; loc. dt., p. 33. 12 j^qq^ ^jt.
558 Scientific Papers of the Bureau of Standards [Voi.ig
(a) VaIvUK oi? Mutuai. Inductance from the Charts.—On chart 3 (fig. 4) the value of -r-t read from the curve -j-=i,
when
AD-x = o.8, IS 6.7
HenceM= 168 millimicrohenrys
The error is less than i per cent.
(b) VAI.UE OF M FROM Tabi.es Using k^ as a Parameter.—By interpolating in Table 4, ^^ = 0.8622. Using this value, the
value of H from Table i is 3.103.
.'. M = rj^ H = 167.1 millimicrohenrys
This result is in error by less than one-tenth of i per cent. Thesame result would have been obtained had a more accurate
value of k^ been used.
(c) Computation from Tabi.es which Use Oni.y the RatioOF THE Dimensions.—Interpolating in Table 6, P = 6.700
.•.M = -T-P = i67.5 milHmicrohenrys
This is in error by a little over two-tenths of i per cent.
(d) Computation by FormuIvA (18) Invoi^ving k^.—Substi-
tuting in equation (18)
M[8.71035 -f-7. 23378 X0.860335]
53.8516
1 .00000 — TT
+ .03448
+ .00030
+ .00001
4.00000
— .06897
+ .00030
+ .00001
= 14.93382X1.03479-3.14159X3.93135
.•.M= 167.082
In this case only four terms were necessary to give the values of
the series to within i part in 100,000. However, as the result
is the difference of two quantities which have values nearly
alike, the result is not known with that accuracy. This com-
puted result differs from the correct result by less than 3 parts
in 100,000.
^pa^s] Mutual Inductance of Coaxial Circles 559
(e) Computation by a Formui^a which Invoi^vbs Only theMeasured Dimensions.—Equation (19) is suitable for this com-
putation.
M = 27r^ • 25 • [0.67703P • 1.00000
+ .07879
+ .01035
+ .00158
+ .00026
+ .00005
+ .00001
= 493.48 X0.31033 X 1.09104
= 167.085
This result is correct to within i part in 100,000. The conver-
gence of the series is such that no difficulty would be experienced
in obtaining a higher accuracy if desired.
EXAMPLE 2 ,
Measured dimensions:
A =a = 25 cm D = i cmComputed constants:
r\ = 2,soi ri = 50.0100
r\=i r, = i.
y2^'2 =-2? =0.000399840
Rosa and Grover value of mutual inductance is 1036.666 milli-
microhenrys.
M(a) Value from Charts.—From the third chart (fig. 4), -j
is approximately 40, but the curve is changing so rapidly at this
point that the value is little more than an estimate. This gives
M = 1,000 millimicrohenrys
which is in error by nearly 4 per cent.
(b) Value from Table in k'^.—^With coils so close together
it is impossible to obtain k'^ with sufficient accuracy from Table
4. Using the value of k'^ given above
colog ^'^ = 3.3981
From Table 2 // = 20. 728
.-. M = 1,036.6
This is m error by less than i part in 10,000.
4-00000— .00020
560 Scientific Papers of the Bureau of Standards [Vd.io
(c) Computation by Tabi.es in ^ and -j—Table 6 can not
be used when the coils are so close together.
{d) Computation by a FormuIvA Invoi^ving k'^.—^Using for-
mula (18)
M r„, n o T I
1.00000 —--|_8.7io35 + 7.23378 X3.39811J
I^ ^^^
= 33.29489 X 1.0001 —ttX 3.99980
.•.M = 1,036.665
This formula is especially well suited to this example. Only-
two terms are required in the series, yet the result is accurate
to I part in 1,000,000.
{e) Computation from Measured Dimensions.—^For these
constants, equation (19) is not sufficiently convergent to makethe computation practicable.
EXAMPLE 3
Measured dimensions:
A=a = io D = ioo
Computed constants
:
r\ = 10,400 ^1 = 101.980
^' = 0.0384615
a DA = ' Z=^^
Rosa and Grover value of mutual inductance is o. 1916496 milli-
microhenrys.
(a) Vai^ue of Chart.—^Value of -j beyond range of chart.
(6) Vai^ue from TabIvE in k"^.—^With coils so far apart, it is
impossible to obtain k"^ with sufficient accuracy from Table 5.
Using value of k^ computed above
^* = 0.001479
From Table 3 H = 0.00188
1
.•.M= 0.19 18 millimicrohenry
This is in error less than one-tenth of i per cent.
(c) Values from Tabi.e Using Ratio of Dimensions.—From Table 8
5=19.18
.•.M =—==v— • 19.18100^
= 0.1918 millimicrohenry
The error is less than a tenth of i per cent.
sTjks] Mutual Inductance of Coaxial Circles 561
(d) Computation by a Formui.a in k^.—By equation (17)
M— =1.23370x0.001479 29 1.00000^1 + .02885
+ .00087
+ .00003
M= o.i9i65o
The error is less than i part in 100,000.
(e) Computation by a Formula InvoIvVing MeasuredDimensions.—Equation (19) is suitable for this example.
M= 2ox^ [0-09901 95P I .000000
+ .000036= 0.191649
This result is in error by only 2 parts in 1,000,000. It has been
carried to this accirracy to show the ease of computing by this
formula when the coils are far apart.
X. SUMMARYCharts are given from which the mutual inductance of two
coaxial circles can be read. Under favorable conditions the
result will not be in error by as much as i per cent. These charts
are also useful for designers.
Tables are given by means of which the computation of the
mutual inductance of any two coaxial circles with an accuracy of
0.1 per cent is easily accomplished. These tables can also be
used for the inverse problem of finding two circles which will give
a desired inductance.
Formulas are given by means of which the mutual inductance
of any two coaxial circles can be computed with an accin*acy of
0.0 1 per cent. The coefficients of the terms in these formulas
are expressed in decimal form, and they are so arranged that the
labor of computation is reduced to a minimum. When the coils
are either close together or far apart, these formulas can be used
to give an accuracy even greater than 0.0 1 per cent. In the
intermediate range the number of terms required to give an
accuracy of even i part in 100,000 is very large. For this range,
Nagaoka's formula ^^ is very convenient, especially if the tables of
Nagaoka and Sakurai^^ of k^ are available.
The authors wish to acknowledge assistance from a number of
their associates. In particular. Dr. Chester Snow has aided in
simpHfying the mathematical expressions.
Washington, April 14, 1924.
w Given as equation (8) in Rosa and Grover's paper.
APPENDIX
TABLES AND CHARTS FOR DETERMINING THE MUTUAL INDUC-TANCE OF COAXIAL CIRCLES
Two types of tables are given in this appendix, namely, single entry tables anddouble entry tables. In the single entry tables the parameter (k^, colog k^^, or k*)
is a somewhat complicated function of the measured dimensions (the radii of the
circles and the distance between their centers). In the double entry tables each
parameter is either the ratio of two measured dimensions or the square of such a
ratio. Hence the parameters for the double entry tables are much more easily
determined than are those for the single entry tables. However, the interpolation
in the double entry tables is more diflBcult than with the single entry tables.
So far as possible the tables have been constructed so as to give an accuracy of
one-tenth of i per cent in the value of the computed mutual inductance. However,
when it is necessary to determine the inductance with this accuracy, it is suggested
that computations be made by both types of tables. Should they fail to check with
the required accuracy a third computation should be made, using one of the formulas
given in the text.
TABLE 1.—Values of H in the Equation M=ri H
Method of Using Table to Compute an Inductance.—Given A and a, the radii
of two coaxial circles, and D, the distance between their centers. Compute
and
rj=V(A+a)2+D2
If the value of k"^ lies between o.i and 0.85, obtain the value ofH from this table, using
linear interpolation if necessary. Then M=ri H.
Method of Using Table to Find Size and Position of Circles which Will
Give a Desired Inductance.—Choose a value for A. Select a value of H near the
Mvalue given by —j, for which a value of h? is given in the table. Then
M
a=—~4A
D=^r\-{A+af
MIf D is imaginary, the value of H has been taken too far from the value given by —j*
Units.—Values of ^ , a, and D are in centimeters. Value ofM in millimicrohenrys.
li A, a, and D are in inches, multiply the value of // by 2.54 to give M in milli-
microhenrys (thousandths of a microhenry).
Accuracy.—For values oi k"^ between o.io and 0.85, values of H accurate to one-
tenth of I per cent can be obtained by linear interpolation.
562
Curtis -]
Sparks] Mutual Inductance of Coaxial Circles 563
k' H Difference k'' k' H Difference k''
0. 0001243 1.00 0.50 0.5016 0. 0271 0.50.01 .0001243 . 0003767 .99 .51 .5287 .0284 .49.02 .0005010 .000635 .98 .52 .5571 .0295 .48.03 .001136 .000899 .97 .53 .5866 .0309 .47.04 .002035 . 001170 .96 .54 .6175 .0321 .46
.05 . 003205 .001446 .95 .55 .6496 .0336 .45
.06 .004651 . 001730 .94 .56 .6832 .0350 .44
.07 .006381 . 002020 .93 .57 .7182 .0366 .43
.08 . 008401 .00232 .92 .58 .7548 .0381 .42
.09 .01072 .00262 .91 .59 .7929 .0399 .41
.10 .01334 .00294 .90 .60 .8328 .0416 .40
.11 .01628 .00325 .89 .61 .8744 .0435 .39
.12 .01953 .00358 .88 .62 .9179 .0455 .38
.13 .02311 .00392 .87 .63 .9634 .0476 .37
.14 .02703 .00427 .86 .64 1. 0110 .050 .36
.15 .03130 .00462 .85 .65^
1. 061 .052 .35.16 .03592 .00498 .84 .66 1.113 .055 .34.17 .04090 .00536 .83 .67 1.168 .057 .33.18 .04626 J .00574 .82 .68 1.225 .060 .32.19 .05200 .00613 .81 .69 1.285 .064 .31
.20 .05813 .00654 .80 .70 1.349 .066 .30
.21 .06467 .00696 .79 .71 1.415 .070 .29
.22 .07163 .00739 .78 .72 1.485 .074 .28
.23 .07902 .00782 .77 .73 1.559 .078 .27
.24 .08684 .00828 .76 .74 1.637 .082 .26
.25 .09512 .00878 .75 .75 1.719 .087 .25
.26 .1039 .0092 .74 .76 1.806 .092 .24
.27 .1131 .0097 .73 .77 1.898 .097 .23
.28 .1228 .0102 .72 .78 1.995 .104 .22
.29 .1330 .0108 .71 .79 2.099 .111 .21
.30 .1438 .0113 .70 .80 2.210 .118 .20
.31 .1551 .0119 .69 .81 2.328 .126 .19
.32 .1670 .0124 .68 .82 2.454 .136 .18
.33 .1794 .0130 .67 .83 2.590 .146 .17
.34 .1924 .0137 .66 .84 2.736 .158 .16
.35 .2061 .0142 .65 .85 2.894 .171 .15
.36 .2203 .0150 .64 .86 3.065 .187 .14
.37 .2353 .0156 .63 .87 3.252 .205 .13
.38 .2509 .0163 .62 .88 3.457 .227 .12
.39 .2672 .0171 .61.89 3.684 .251 .11
.40 .2843 .0178 .60 .90 3.935 .283 .10
.41 .3021 .0186 .59 .91 4.218 .321 .09
.42 .3207 .0194 .58 .92 4.539 .369 .08
.43 .3401 .0203 .57
.44 .3604 .0211 .56 .93 4.908 .433 .07.y4 5.341 .520 .06
.45 .3815 .0220 .55 .95 5.861 .647 .05
.46 .4035 .0230 .54 .96 6.508 .848 .04
.47 .4265 .0240 .53
.48 .4505 .0250 .52 .97 7.356 1.215 .03
.49 .4755 .0261 .51 .98 8.571 2.114 .02
.50 .5016 .0271 .50 .991.00
10. 68500
00 .010.
564 Scientific Papers of the Bureau of Standards [Vol. IQ
TABLE 2.—Values of H in the Equation M=>-i H When the Circles are Close
Together (^2;>o.85)
Method of Using Table to Compute an Inductance.—Compute k ^ and r ^ as in
Table i . Obtain k^^ from the relation k^^= 1 —k^, or compute from the equation
k'^--
Find, colog ^^^. Use linear interpolation to get value of H from table. Then M=ri H.Method op Using Table to Find Size and Position op Circles Whick Will
Give a Desired Inductance.—Method identical with that given under Table i.
Use this table only when necessary to interpolate betvv'een values given in Table i.
Units.—A, a, and D in centimeters; M in millimicrohenrys. If A, a, and D are in
inches, multiply the value of // by 2.54 to giveM in millimicroheniys (thousandtlis of
a microhenry).
Accuracy.—Throughout the table, an acctuacy of one-tenth of i per cent can be
obtained by linear interpolation. For values of colog h/^ greater than 2 .0, an accuracy
of one one-hundredth of i per cent can be obtained by linear interpolation.
Colog H Differ-ence k'' V Colog H Differ-
ence k'' k*
0.8 2.759 675 0.1585 0. 8415 2.5 14. 255 718 0. 003162 0.9968.9 3.334 601 .1259 .8741 2.6 14. 973 719 . 002512 .99751.0 3.935 623 .1000 .9000 2.7 15. 692 720 .001995 .99801.1 4.558 641 . 07943 .92061.2 5. 199 655 . 06310 .9369 2.8 15. 413 721 . 001585 .9984
2.9 17. 133 720 . 001259 .99871.3 5.854 668 .05012 .9799 3.0 17. 855 722 .001000 .99901.4 6.522 676 . 03981 .9602 3.1 18. 576 721 . 0007943 .99921.5 7.198 687 . 03162 .9684 3.2 19. 298 722 . 0006310 .99941.6 7.885 693 . 02512 .97491.7 8.578 698 . 01995 .9801 3.3 20. 020 722 . 0005012 .9995
3.4 20. 742 723 . 0003981 .99961.8 9.276 703 .01585 .9842 3.5 21. 465 723 . 0003152 .9997L9 9.979 706 , 01259 .98/4 3.6 22. 188 723 .0002512 .99972.0 10. 685 710 . 01000 .9900
2.1 11. 395 712 .007943 .9921 3.7 22.911 723 . 0001995 .99982.2 12. 107 714 . 006310 .9937 3.8 23. 6.34 723 . 0001585 .9998
3.9 24. 357 723 . 0001259 .99992.3 12. 821 716 .005012 .9950
;4.0 25. 080 , 0001000 .9999
2.4 13.537 718 .003981 .99601
Curtis 1Sparks! Mutual Inductance of Coaxial Circles 565
TABLE 3.—Values of H in the Equation M^r^H When the Circles Are FarApart (i^2<0.1)
Method of Using Table to Compute an Inductance.—Compute k^ and r^ as in
Table i . Obtain k^. Use linear interpolation to get value of H. Then M= r j H.Method op Using Table to Obtain Size and Position op Circles Which Give a
Desired Inductance.—Method identical with that given under Table i. Use this
table only when necessary to interpolate between values given in Table i.
Units.—A, a, and D in centimeters; in millimicrohenrys. If A, a, and D are in
inches, multiply the value of H by 2.54 to give M in millimicrohenrys (thousandths
of a microhenry).
Accuracy.—Throughout the range of this table, H is nearly a linear function of
k^. However, as both k^ and H start at zero, the percentage error in H may be large
when k"^ is very small. Hence when k^ is less than -o.ooi, it is better to computethe mutual inductance by the formula
M=i.2337 r^fe* (i+K^2)
This formula will give an accuracy greater than one-tenth of i per cent when k'^ is
less than o.ooi. When k^ is greater than o.ooi and less than o.oio, an accuracy
greater than one-tenth of i per cent can be obtained by the use of the table.
«* H Differ-ence
*« V» k' H Differ-ence V k''
.001
.002
.003
.004
.005
. 001264
.002553
.003860
. 005181
.006515
1,2641,2891,3071,3211,3341,345
.03162
.04472
. 05477
. 06325
. 07071
1.000.9684.9553.9452.9368.9297
0.006.007.008.009.010
0. 007860.009216. 010582. 011957.013341
1,3561,3661,3751,384
0. 07746.08367.08944.09467.10000
0.9225.9163.9106.9053.9000
566 Scientific Papers of the Bureau of Standards {Vol. 19
I
o
g
73 .tj '5
" ^ S
JH ^ ^ CO
H
«fl OT rj
2 2 §
"S 'S .^
II II II
-. a
OS
o(Upi
Sil QK
Oh
Q
T3
Oa
'd
Ia
I
Qk§
II ^
I
B
ft
6
o
0000
f̂j6
.2 XCN
< 6-d 1
rt roc3 M
HO)
dX
S310
t« 61
5 rt
^ O^ ^
I
+ '5
Qk
4-. fl
z §o y
nS O
Si
O fH
3 ^
«N
ft ^
-^^"sQ
g,
fr -t?
Qk
*d o
6 -d
^ So
CO ^
.2'^
ft (U(U
'ft
«H-^ 6
CO O
3^5
llft-^
i.iy to< O
Curtis 1Sparks!
Mutual IndtLctance of Coaxial Circles 567
: I*
I
r>< 0» CM »< »H©JO>*5?M^O»-i a>oooo>0^
I
0«I
0\I
00I
ssss
«5 §5 is :
i i
^ isj
s i^ i ^ is i g is is
; !
2 Iflf STSoT iflfSS5|S|
«s S i§ i S is i 2 :i i §isi I|s|l
0.8 iflf iflf
-00 !<o 1
2 IS ; Sjii i i M teo '
e<5 I<M loj
d ^^1 <^ 1
ifi« ifgs t^1t^
1t^
•o s s iiij § i5i §i§i S ^ 1
S ". 1 "»1ifi? "T"" ^ 1 ®T^.
•0I(N 1
sili §ili tiii iiiii
in
6 ifegf
K> s is i s Is i S is i
l> It- 1 gjii i ig Ist- It— It—
dSfeSS o^inoo
s"f!S7S
<« 11 11 11j^ I
!S ! iji 1
sagss"1 l*^ 1 •^ l-^ 1
§SiSI
«s
10 I® 1
s is i 3 i3 i 1 :l : 1 III iiiii
d ^,1 '^ 1
™ ,<o (to
»o
1(N 1
(M 155 I iiil is S ii i iilil
d iiiS-! 1 ^.
1
ssjij S 1 S 1 s
r— io>
ii§
!°'/
Qj^
o 'd c> <J o <d o <] o <] o < o <o <ir-;
568 Scientific Papers of the Bureau of Standards Wol.io
-^ 2-2i
to ^rt a
• »-i
to
1(U u
^1
II8(U ^ CO
« > "^
o
Ij
< (L)
(U ^r1 to
1u (U
si4->
w bfl PI ti
^ •S Si«.t3
O s ^14 a Sia ^ *S 8dji S-i p^ MH ^ 2 °u ^ ;3 <y
<u (33
i
1S^1
to'y-l o fii
(U
1to
5|^ o
->
o ^ "iCO aa3
t;! •^ «
=?
O.tJ6
o <o a1
1 ^1Pl^ S
to O
5 3^< ^H
1
o .
1<
o1.5^
i? rt 1=1
hJ 7 ^
8
^ g^^ 8 ^s&fe^ I'Z U
2fS57
.28571687
.44441011
.5455
699.6154
613.6667
392o^^-vo~<*o
I
-I In I
I 1?5 I iil!(M lo 1
8 :sI
Sill a !§ is
do
IIP t- c- t-
«s INOT It- 1 ^ I2 i
CO \coj
i'iijiisii
so
CO Tt< PIS K-R-R
-1 lie 1
i i^ iliij i il
1
t- lo I
iiiii
do
PP co^sooooo
•o! i^ :
siSi S is i IS s is is
So
PP PP <yiO5 00-^ro
•o Nsi Sill S lis 1 §iii i|i|i
2o
Pll PPX5 jiij sjil ^ ;§ i
«*< I>o 1
50 It^ 1 g 1 g
d IISo
* lO >o
<«' iM I ii S3 \s ;
r- Itj* I
§8 ISB 1
g is i^05 '05 'Ci
2o
PP TjtOONSO >n95oo>^
«si is i
1 leo 1 IS S is ;
00 io> '
O Io3 1
o lo I 2 l§ Is
do
PP M '^CO
'"'CO
•«
1 '<C^ 1
I;« i
ip S :g i
i'iSi^ i« iseo 1'* iTj<
do
PP PP PP »ot-co*-o
"O i|Sj il00 leo 1
1 iS i s:i; S li:; IS
o O jO I o jo 1 o jo 1 o jo I o jo jo
o<id<iro IfO I
d<Jd<l d<ld<id tr-! I 00 I e?I id
Curtis 1Sparks} Muttuil Inductance of Coaxial Circles 569
OH
Iao
^
•3 g
V ^
.in y
bo ci
^ B
66
2 H ^
(U q; (U
11 II II
a « a>
<u
" rn
'*^ s ^
-d g ^6
>^6|t3 ^ a; CO
4> as _co ^
4^ 'W .y rt
-^6^6>. o
-d ^ a os
si
;^ ^^^.2 ^ -
• » ^^^3 J3 ««QiH6Q
o § <w ^r
^ .t ^ >.
a « ;^^
>^'d
I '^
q2
« II +II JTi^
-^ « -4-
« I -
-^^ II
~5- e"a.
2<&
06 li^II I ^
(0 i-J ^ ^
5 J?6|
ir.2.
2 "^ <i>'"6«4=; O O ^
&x ^^
^ :?§
«
s^ ? ^^ H -d {^
6 <» -5 ti
^ II 6 g
^ 6 S3
g o,.2
-^ -d ^ g
S -g §
^ II'i.bo
.J2 g ^ -^
< ?^ ^
^CO J
- II ^-g
§ § g g.
i ^ §P eg (u
fi o o<U H Pi
^ § ««S *-( <u
TT 'd 22
a 3
g° ^ •> b <u*r,
^OT u "d
fill
r! QJ OT
pi TS o
> rt <u .
-d o-^-g
•d 6 g-^
<U tw ^ to
-22 ^ i
^ ? rt ^!^ <U cJ aJ
8 °«6i;
P CO OJ _C2
S ^ -d .2
-d (u g g
3 " >63"^ 6 -2 ^'cS O (L> ^3
I ll -I J -s
&^ 6 -d
fl ii 3 -^
§5 § i
§ i
570 Scientific Papers of the Bureau of Standards [Vol. 10
«.J<
(Ctt> oo s
i 1^ 0)
w :3
.a
1ro oW (L>4
Jli
P^ n<!:{ ^
o
t3
to
s
S"<5
TJ
K iai 'CQ ^1 18 1
I 1<! VB 3O
*̂3Cfl CO
t) 0)
t=.rt
O ^Q Ut
O 9j
?« J:{
o
I>
I
I5
:^fa.
^l<3 "^
Q̂o
B
sPI
^ &TJ <4-l o;
<U o ^-M n-1 H) !h
1a
8 > rt
<i>VM
i ?"o
§ ii ^as
•+3
TJ
1 1y"TS Q
1 c3 45o _ o(U
O oa ^ *a
o •^ ;=!
t^ ^ a^
rt
s .^• IH
o*^ ^s 1.
m ^
Is oP
CO I pa I <vj
KO I uS i Ti' I -^ I CO
OOOO 00 i
2 I 2 I
I vd I
^ 00 VO--HIO»n<tooo"©co
1^- I
;:: {Q Oicgoo Tj<i^
O\-~i00O> O\90VOOit^Of^cooi CO so o is. c^
0^':ocOX5J
I ; I
CslOo rj< <3«IrtOUTO
led I
^O 8s CT( «
I vd I
rJ<90i-f~.O»^?-oaot^
lui
I 1
T}<i^t^oo 0^ O 00 ~*WOiO^OO OOOfMt^^ I O I C> I ON I
CM eg c>j fi
O?^ro*^•^ oouj aolOCOOO^O00* I ts I
voeocoeo o*^co^«soo•-I OS cq U3 r~ »~i m oo vo
t^ i «d I vd I uS I iri
IcD I
00 Tt< •-!^ o<5o<o CO eo o OSo>o«o-4)i^CvIb~ VO'teOO'^O CO ^
t>e4«noot«.Ot^ CO ~*-oo
iod I It^ I I.H I
Tl<OOVO~*-ico»n«is <yi ^sco^o I -^co -* o> ~-i->* ~*o
I «ri I uS
oitoTiooCO OS CO ®4co>«r>.9000 >ooa lo
00 I QC> I
.-4U5VD>-^<*< o cooo
inOOt^i^ Ot-C003T}<VO »0 0( so VO~^Tl<OST»<^>.-<^c^J^=^ oo •~=^<i< so ovd I vd I >o I vri I uS
cot-voo vogjpot-.t-<ootMCo voeocoooo»oc<iui voio-^uaON ' ON I 00 I 00 I
t-» -<^CO '»« J-HOr-tOOCqvo~*cg~* eo-<i-^Qoovd I vd I lo I u-> I ui
^ 1^
CO t^ vo -*
ON I ON I
VO ~*C0 ~*- 00 ~*Tl< so ovd I vd I uS I «r> I vri
8 is ! S is . ^ ... . ^ ...^ .- .^ ,.,
o<o< o<o<] c><]c)< o<o*^ c><3o<3o
Curtis 1Sparks] Mutual Inductance of Coaxial Circles 571
o I o< 1
vn »< CO ^^»o^-.>c>c^^-J^ t>. ~*co so <yi
eo^eo 00-
50 0\»<S
I o^ I 06 I
Tt< -+ai -<<-m
00<© CO
CTl^OCO'O t>~*CO»5CTlvd I vd I in I in I
t~ "O tM 00 «il»
^ 1 to ir-l
ooi^T-ie* a\oio»^fO
I « I I in I in1^ I
I ; I I ; I
o»oin<3i voiwro««^JSOO^O 000006~
»n«SOi^ CO O <V1 "CJ g;(M <£) vo lo oxsm^^oC3 I a\ I cK I 00 I 06
O t^ CO to VO «5 <-l ~*VO
00 < r>^ I vd I vd •
I : I
I ^ I
t«- <o "-H ^ m 'd CT> ~*in
O 1 C> I «3? I 00 I 00
mo*po»^ vo«»tJ<©?po
I t^ I «5 I
I 2 I
I ; I
CO'dOOCO tMOO'^lO CTv ~*m «» ro0> to C-J us t~ to O Ci OEOt^OOCTlt-ooc?>?^ .-itom'o <yi "o CO -<^oo
I 00 I
in 05 in*-.-*CO U5 t~»o<Mt^ I vd I ^
i g; 1 1 7
1 i
1 ;
oowino ineotMoo
inVco 1 CO I csj I
Tj< us CTl X5 Tl<oes voooooCSJ US VO ~*r-(
c> I e3\ I o\
I ~*-co
'
I 00 CO *^
; I od I
r~>-.voo lO-^into cftus-*^*00 »} vo O vo O vo -^ Tj<-^OOoOOOCS .-cOOCOt^ VDtoOUSin I Ti< I CO I
O 'iCJnus ^po so 0000 oTl< us 00 -^T)*
voust-ieoooCO s»o ~*»n00 to CM us vo
I : I I : I I ; I
lOto^Qo cva j^ T»<
«
I CM I 1 ^ 1
^ us vo e« T«<
inuso-*inO I O I ON
vo us rt s« ooeoinus(inoot>*>. ONtocous(ON I 00 I t^ 1 e»: I I
«D iTj" 1 CO"^ '7
; TI : I
.^»s^>-< ooO'~H ON---<ooj^ i-i>-.ooi^Tf «!<! CM »s C7^ us Tj> 00 in^COto t~- '5S! vn t- t^005^00 (M^^mto ootocMus vou^^~J^^oinl'*l colrol ,-;l,-;l cJ'oloN
.-IO.-I--1 ousmus<t~USfMI»S e>9su-)>o<>ooooos^ Oto'<*<us(
ON I 00 I 00 I t> I 1
I : I I ; I
1^ I
^i^Tj««l <N>90ON«« t-i^OJ^COT-(COt^t^ Q*-<0Oto Psl -"-i »^ *^ COcoe^mto CTitoP<ius r- us cm ~*r~(tjIcmI .-;I^1 olcjIoN
coooin»» cotot^oooNco-=^oo»s meoi-Husmt~oooo*^ -^toinuscjN
ON ' 00 I 00 I t-^ I id
53 :SI : I
in'~^Tj«»^ t- eo Tt< 00 vO'-Hincis intoONtoco^^oscMO —leoooto .-c ~, o to rj< ^ ro *^ mooo^oo cof^mto ONtocous t^uspj ~»-t^
inl^l poIcmI ,^1,-jI oIo'Ion
C0i^VO<&?irt ~*o eot^OOONt^
Tl^^^^»Q0ONt^eocoust^»H tom usON
od I t^ I \d
c5<o< o<3o< o<d< o<3o<c>
572 Scientific Pabers of the Bureau of Standards [Vol, 19
a
II
do
Ia
I
liv--
8 Ji I ^!^ a,
1.^0) ^ a
"Z
Sd *$
SI^1
i >
1
1 ° i •43
<u .5 1 i
o 3
1S| 1^
^ o o JH
a (u • i^
t-H
il !
Qa tJ <; U4
h-t
1!o
i
o4> rt O •—O g s .§
rsp ^ i3
sa
So 23
•^ -^ ^»Q be
** c« 5
2 -"
fl o -
O O
Q S
« Q
8 -«
O -qH
^ s
I <u c^J
«a8^
<
CwHslSparks} Mutual Inductance of Coaxial Circles 573
«-i«0'^'o q>c>ono o\o>cO'-« oNoot^oo«noorr^r? o^^S »-iooooo oo<us«o>o^'(•"•St^'O r-190'^90 t^»«0»« <M9»«O«J00CJ* t-S.-J rJcNJ CNIC^N
1-H lO
I : I
5J :S
txosvor* sssscJ <vi cm'
aoa>
^ ^ ^'
;> ; ^
9 :Si : I
t^ '00
I : I
»!< 80 00 »S
«M 80vn«l
cvi c>i ci
O «5 CO oo iH »<s vot^ciOtoONao«MOi<^<to ooao<M>-^co
I ; I ; I
I : I
QOO0^«t00»3i-l«O
l«*VOOO *KSO80C»5
I : I I ! I I : I
<naoao«ocoj^O<to*oloaoooooiiC<4 Cvj CO iif \ri vi
ooo ooomaoooso
oo«»Ot^^cooo«ipg aoioeo ON ao CM so
CO 00 r^ OO ON <S» »HC)»H<r~ooo"»Tj<ao oooooi<M «0 00 VS Tl< KS 0»«ST»»-
CM cq «
I ; I I : I
ON QO CO QO
Psl CO CO
Tt* «i VO I5<t 00 US coa»<Mooot>-eooo>ON»o uT-^t^oovoC0^0«5tn»0 .-H X5 VO «*.-(
CO Ti5 * »o uS NO
«* 1 tC i«.-1 1 « leo
I : I ; I
CM ^*VOO \0 X3 —< 9»pqtooooo \o°o\naomaoooao cvjaoNoao
l>O00i^ ir»<to»-<U5VOI t- O to t-. 'O CO ^*t»
i<©t^>o co»<sooxsco
r-l CM CM
t» l05
I ; I
I ; I
OO^tOrr ^fc-00!5 CQOSCM~c COOJIMOS^<^£^<3^ OlOiOOO> OOOOt^OO IO<toCM>O00xneoONeo cmsonoso O90-<i<w oooofMaoio
»-< >^»ft 05 tH <»00 00 voeo ooiVOCOCM COON «s
O CO VOC) vo
m«i o Ks«n
lO <d NO
I ; I
!3^::r''>2? t^3*03 maoooNo coqovowon<222:^*^ rjOCMO coocooj co2o>-<»^oom so ON -^ CO ~*t>. ~* .-H ^t«o so ON 90 CO ao vo
c>* rHi-! cMCNj oicom>t^cocoo<»
tri <o vd
I
S !g :2
coaovo«» oooovooo -^oscotoON'd'OivO©* OO-^OO ^Cft.-(000>CO -^t^ -* rH -^t-NO •>* O 00 Tj< 90 t^
CM ;<M ; CO ;fo ; ri< [t»<
o<3<3< d<c>< o<o< c><1c><3o«rt
JNO [ t» I
00ION
Io
o<lc)<d<l c><c><i-J
574 Scientific Papers of the Bureau of Standards [Vol. IQ
t:
<u
10
II
:^
ao
cr
> g
« d< iH 1^
rt ^
<L> ^
td -25
^
^^ E to
^ •otQ
i> 0)
.2 .^?J
tort
13
:a>•
?H T* (U<^
sd 3 >%
•3Ci3
.1*?3 3
*© to ;j
Q. 4) Q;h
11
1^ 1,
.t3
•d w rt4) .^3
o ^ lU
11H %
g Qfl U P tj'^ d) ^
§(U P.HH
<3
< (0
gi &• J-( •5 -^ (UO p
Ȥ
g^ ?b .§C3
'•A 1^12 e
a Co H« J A '^
o ^ ^ ^
i
5'B
<u 2; o <v
|1> B
S5i
1^a
to "tirt % f2 oM < N* 'd
i;i toto H
en
It+j B
^ a 1 .sla^ fe^ 'cS
qt<u ^ o ?r-?*
I& 1to
« 2- o
21§2
o81to
s2; ^ a o ;3
.c °
1 rt
'-5
>• 11>; o
O ^
W ^8-H -sC 'So
la^ ? s « B
1 1 ^
Curtis 1Sparks]
Mutual Inductance of Coaxial Circles 575
19.
739llli
18.
625
17.632
888
16.
744802
uS in
(NJ^VOOOto o ^ ^
S 2 2 :^ ;:3
•o<=> is i S i§ S ig i S io 1
2 'S '52
°2o o;'~'od 2 id
Si2&CO CO PJ
*«° JS i § is s is i sjii 8 :i il
St>oi>os2^2
Sil3 Sisi2 id 2 2 2
««° js i » jCO 2 is i
o lo !
§ § is
o-0( ^^ i-i «~.
O »-i ON <3i
2 2ssss
S 2 2
»o® is
1
s is 2 :2 i iili ijip
2S T 2 2 a 3
§515S«i" CO CO
•o° 1"
:
§ ;g S i| i s \i \ § ii i§
22 2 00 t>: 2 "^ s s s
•o° js i § jl 2 i§ i a Ii iigii
22 S
OOXiCOXS
2 2 2 2 * * CO
•o=> is
I
§ ig 2 :2 I li§i 1 i§ is
S2 S 00 t^
55 ^ w> to ass»o °
I" I
s ii S ;2 I a ii i ajiis
o2 2 00 !>: ^ a ^ s s
<o= is i 2 is Sill ^jii § ig ii
5gill2 2 2 2
SSSSs«n uS tjJ
•o ° jisi 2 Is g il ; 1 i l|gii
o2 2
SSg52oo-<fcoaoouS >n uS
1iS :
s isc><]c><
§ :2 1
o<1c5<d is i
c><o<12 is iso <3 o < d
576 Scientifc Papers of the Bureau of Standards [Vol, IQ
TABLE 9.—Values of2^-(|n)4
The above expression appears in all the formulas for mutual inductance given in
this paper. By the use of this table, the series can be easily extended to the twenty-
sixth term. Let
(J2«)2 ^
n log ^+10 4^ n log ^+10 ^
10. 0000009. 3979409. 1480628.9897008. 8737188. 782202
8. 7066268. 6422528.5861968. 5365508. 491998
8. 4515908.414624
1. 00000000.2500000
. 1406250
.0976563
. 0747684
.0605623
.0508892
. 0438785
.0385652
. 0343993
.0310455
.0282872
.0259791
13 _ 8.3805588. 3489708.3195208. 2919448. 2660148. 241546
8. 2183808. 1963908. 1754608. 1554908. 136404
8. 1181148.100566
0. 024019214 . 022334215_ . 020869916 . 019585917 . 018450718 .0174400
19 . 016534120 _ . 015717721. . 014978222. . 0143051
10__ 23 . 0136900
11 24 . 013125412__ 25. .0126057