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

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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