⬇ Download Markdown · ⬇ Download LaTeX · ⬇ original · view original
— .
FORMULAS AND TABLES FOR THE CALCULATION OF MUTUAL AND SELF-INDUCTANCE
By Eld ward B. Rosa and Frederick W. Grover
CONTENTS
(See also Index)
Page
Introduction 5 i. Mutual Inductance; of Two Coaxial Circles: (a) Formulas— I, 2, Maxwell's formulas in elliptic integrals 6, 7 3, Formulas for the elliptic integrals and F E 8 4, Maxwell's formula in terms of and Q F E 9 5, Maxwell's formula expressed as a k series 9 6, Maxwell's formula expressed as series in k x 10 7, Weinstein's k f series formula 10 8, 9, Nagaoka's q series formulas 11 10, 11, Maxwell's series formulas for any two coaxial circles 13 12, Maxwell's series formula for circles of equal radii 13 13, Coffin's extension of (12) 14 14, Series formula for any two circles, extension of (10) 14 15, Formula for coplanar circles, derived from ( 14) 14 16, Havelock's formula (extended) for circles near together. 16 17, Havelock's formula, for circles far apart 16 18, Mathy's formula (corrected) 17 19, Simple form of Mathy's formula for the case ra 2 =2 r2 2 . 18 {b\ Choice of formulas , 19 (c) Summary of formulas . . „ 19 (d) Examples 1-19, illustrating the formulas for the mutual inductance of coaxial circles , . 20 2. Mutual Inductance oe Two Coaxial Coils: (a) Formulas 20, Rowland's formula, equal radii, but unequal section .... 33 21, Rowland's formula, equal radii and equal section 33 22, Differential coefficients in (20) and (21) 34 23, Rayleigh's formula of quadratures 34 24, Same for coils of equal radii and equal section 35 25, 26, Formulas for errors of formulas (21) and (24) 36, 37 27, 28, Lyle's formulas 38 29- 31, Rosa's formulas for two equal coils 39 32, Same for coils of equal radii, but unequal sections 39 33, The Rosa-Weinstein formula 40 34, Formula in terms of self-inductance 41 35, Geometric mean-distance formula 42 (b) Choice of formulas 43 (c) Examples 20-33, illustrating the formulas for the mutual inductance of two coaxial coils 44 1 —— —
2 Bulletin of the Bureau of Standards Woi. 8, No. i
Mutual Inductance of Coaxial Solenoids: (a) Formulas Page 36, 37, Maxwell's formulas for two concentric solenoids of equal length , 53, 55 38, Havelock's formula for concentric solenoids of equal length , 56 m
39, Roiti's formula, inner solenoid the shorter, coils con-
centric 57
40, Gray's formula, coaxial coils not concentric 59
41, Formula for the Gray electrodynamometer 60
42, Searle and Airey's formula, inner coil the shorter, coils
concentric 61
43, Rosa's extension of (42) 62
44, Cohen's absolute formula in elliptic integrals, coils
concentric 64
45, Nagaoka's absolute formula for any coaxial solenoids. 65
. .
46, 47, Russell's formulas 67, 68
(6) Mutual inductance of a short secondary, on the outside of a long primary 68 (c) Formulas for single layer coils of equal radii 48, Rosa's formula 70 49, Rosa's formula, coils of equal breadth 70 50, Rosa-Weinstein formula 71 51, 52, Mutual inductance by means of self -inductance formulas. 72 (d) Other formulas and Olshausen's absolute formula 73 (e) Choice of formulas 73 (/) Examples 34-47, illustrating the formulas for the mutual induc- tance of coaxial solenoids 77
Mutual Inductance of a Circle and a Coaxial Single-Layer Coil: (a) Formulas 53, Iyorenz's series formula 98 54, Jones's formula in elliptic integrals 99 55, Campbell's form of (54) 100 56, 57, Rosa's series formulas 101, 102 (6) Examples, 48-51, illustrating the formulas for the mutual induc- tance of a circle and a coaxial single-layer coil 103
The Self-Inductance of a Circular Ring of Circular Section: (a) Formulas 58, 59, Kirchhoff's formulas no 60, 61, Maxwell's formulas no 62, Maxwell's formula when the ring is a hollow, circular thin tube in 63, Rayleigh and Niven's formula in 64, Tube bent into a circle 112 65, Tube of infinitely thin walls 112 66, Wien's formula, current density proportional to the dis- tance from the axis of the ring 113 67, Thomson's formula for a ring of elliptical section 113 (b) Examples 52-56, illustrating the formulas for the self inductance of a circular ring of circular section 114
The Self-Inductance of a Single Layer Coil or Solenoid: (a) Formulas 68, Formula for infinitely long solenoid 116 69, 70, Rayleigh and Niven's formulas 116, 117 71, Coffin's series formula, extension of (69) 117 72, 73, Lorenz's absolute formula in elliptic integrals 118 74, Formula for use with Table IV, derived from (73) 118 75- 78, Nagaoka's q series formulas 119-121 79, Webster-Havelock formula 121 80, Rosa's formula for obtaining L from L s . . 122 8i, Summation formula 123 — .
Grover
Formulas for Mutual and Self-Induct an ce 3
4 Bulletin of the Bureau of Standards [Vol. 8, No. z
Formulas for Geometrical and Arithmetical Mean Distances— Continued. Page 136, G. M. D. of two circular areas from one another 170 137, A. M. D. of a line from itself 171 138, A. M. S. D. of a line from itself. 171 139, 140, A. M. D. and A. M. S. D. of a circular line from itself . 171 141, A. M. D. of an external point from the circumference of a circle 172 142, A. M. D. of an external point from a circular area 172
High-Frequency Formulas and Examples: (a) Formulas 143, 144, Kelvin's formula for the resistance and inductance of straight cylindrical wires 174 1440, Formulas denning IV, F, and Z 174 145-147, Formulas for the absolute and fractional change of in- ductance of straight wires with frequency 174 148, Defining formulas for ber and bei 175 149, Formulas for W, Y, and Z, small argument 176 150, Savidge's asymptotic formulas for W, Y, and Z 176 151, Rayleigh's formulas for straight wires, low frequency. 177 152, 153, Rayleigh's limiting formulas 178 154, 155, Formulas for limiting change of inductance of straight wires 178 156, Formula for the argument x 178 157-160, Resistance and inductance of return circuit of two parallel wires 180 161, 162, Limiting change in inductance of circular ring 181 163-166, Resistance and inductance of circular ring 182 (b) Examples 82-85, illustrating the formulas for inductance and re- sistance at high frequency 183 NOTE: Recent formula for coaxial circles (Nagaoka) 187
APPENDIX
TABLES OF CONSTANTS AND FUNCTIONS USEFUL IN THE CALCULATION OF MUTUAL AND SELF-INDUCTANCE
I. Maxwell's table of JL -\(%-k\) F—$E~\
^4Aa~\\k k J,
for formula (1) . 190
II. Log F and log E as functions of tan 7 193 III. Table of K's for formula 57 193 IV. Table of Q's for formula 74 194 V. Table of constants A and B for formula 82 195 VI. Table of constants^ andy 2 for formula 90 196 VII. Table of constants A for formula 80 197 VIII. Table of constants B for formula 80 199 IX. Table of constants A s for formula 91 200 B X. Table of constants s for formula 91 200 XI. Table of Napierian Logarithms for numbers 1-100 201 XII. Values of Fand E, o° to 90 . , 202 XIII. Values of log .Fand log E, 45 to 90 204 XIV. Table of binomial coefficients for interpolation formula 213 XV. Values of g q and e for use in formulas 8, 9, 45, 76, 77, and 78 } x , 215 XVI. Values of e x and e/ as function of qu for use with formulas 6 and 6a. 218 XVII. Coefficients in hypergeometric series of formula 18 220 XVIII. Location and magnitude of maxima and minima of coefficients in Gray's and Searle and Airey's formulas 221 XIX. Values of coefficients in Gray's and Searle and Airey's formulas 40, 43. 56 222 XX. Nagaoka's table of the correction for the ends, as function of 0, formula 75 223 Groler] Formulas for Mutual and Self-Inductance 5
APPENDIX— Continued
TABLES OE CONSTANTS AND FUNCTIONS, ETC. —continued.
Page
XXI. Nagaoka's table of the correction for the ends as function of diameter 224 length XXII. Values of ^L — -X ^ and A % for use in formulas for high fre- Y 2 Y Y x Y quency 226 XXIII. Values of limiting value of the change of inductance with fre- quency 229 XXIV. Values of argument x in high frequency formulas 230 Index 231
INTRODUCTION
A great many formulas have been given for calculating the mutual and self-inductance of the various cases of electrical circuits occuring in practice. Some of these formulas have subsequently been shown to be wrong, and of those which are correct and appli- cable to any given case there is usually a choice, because of the greater accuracy or greater convenience of one as compared with the others. For the convenience of those having such calculations to make we have brought together in this paper all the formulas with which we are acquainted which are of value in the calculation of mutual and self-inductance, particularly in nonmagnetic circuits where the frequency of the current is low enough to assure sensibly uniform distribution of current. In the last section some formulas are given for the variation of the self-inductance and resistance with the frequency. A considerable number of formulas which have been shown to be unreliable or which have been replaced by others that are less complicated or more accurate have been omitted, although in most cases we have given references to such omitted formulas. Where several formulas are applicable to the same case we have pointed out the especial advantage of each and indicated which one is best adapted to precision work. In the second part of each section of the paper we give a number of examples to illustrate and test the formulas. We have given the work in many cases in full to serve as a guide in such calculations .
in order to make the formulas as useful as possible to students and others not familiar with such calculations, and also to facilitate the work of checking up the results by anyone going over the subject. We have been impressed with the importance of this in reading the work of others. :
6 Bulletin of the Bureau of Standards [Voi.8,jvo.i
In the appendix to the paper are a number of tables that will be found useful in numerical calculations of inductance. In most cases we have given the name of the author of a formula in connection with the formula. This is not merely for the sake of historical interest, or to give proper credit to the authors, but also because we have found it helpful to distinguish in this way the various formulas instead of denoting each merely by a number. The formulas of sections 8 and 9, which are taken largely from a paper by one of the present authors, 1 are, however, not so designated, although the authorship of those that are not new is indicated where known. This paper includes practically all the matter contained in the 1907 paper under the same title by Rosa and Cohen, but in addition to a thorough revision in which some errors are corrected and some formulas extended, a large amount of new matter has been added both in the body of the paper and in the tables. We shall be grate- ful to anyone detecting any errors either in formulas or tables if he communicates the same to us.
1. MUTUAL INDUCTANCE OF TWO COAXIAL CIRCLES
MAXWELL'S FORMULAS IN ELLIPTIC INTEGRALS
The first and most important of the formulas for the mutual in-
ductance of coaxial circles is the formula in elliptic integrals given by Maxwell 2
m= 4 A/C^! - k\f- ~d
7T [l]
in which A and a are the radii of the two circles, d is the distance between their centers, and
k= .
v = sin 7=-*-J 1
^/(A + af + d
2
rx
.Fand E are the complete elliptic integrals of the first and second kind, respectively, to modulus k. Their values may be obtained from the tables of Legendre (see Tables XII and XIII in the Appen- dix), or the values of M+ ^ir^Aa may be obtained from Table I in the appendix of this paper, the values of 7 being the argument.
^his Bulletin, 4, p 301; 1907.
2
Electricity and Magnetism, Vol. II, \ 701.
:
Rosa I Formulas for Mutual and Self-Inductance Grover]
The notation of Maxwell is slightly altered in the above expres-
sions in order to bring it into conformity with the formnlas to follow. Formula (i) is an absolute one, giving the mutual inductance of two coaxial circles of any size at any distance apart. If the two circles have equal or nearly equal radii, and are very near each other, the quantity k will be very nearly equal to unity and 7 will be near to 90 °. Under these circum- stances it may be difficult to obtain a sufficiently exact value of F and E from the tables, as the quantities are varying rapidly and the tabular differences are very large. Under such circum- stances the following formula, also given by Maxwell 2 (derived by means of L,anden's trans- formation), is more suitable
\JS*u>,
A [2]
in F which 1 and Ex are complete elliptic inte- grals to modulus kly and
*,-' sin~" = 4. Aa
7l
^i +r 2 C^ + r,)"
rx and r3 are the greatest and least distances of Fig. 1 one circle from the other (Fig. 1); that is,
r^ ^A + ay + d*
r2 =^(A-ay + d 2
The new modulas k differs from unity more than k, hence y is not x t
so near to 90 ° as 7 and the values of the elliptic integrals can be taken more easily from the tables than when using formula (1) and the modulus k. Another way of avoiding the difficulty when k is nearly unity is to calculate the integrals F and E directly, and thus not use the 8 Bulletin of the Bureau of Standards vol. 8, No. i
tables of elliptic integrals, expanding F and E in terms of the complementary modulus k\ where k! = V 1 — k % > k' may usually be r more accurately calculated by the formula k = — ! . The expressions ri for F and E are very convergent when k! is small.
2
2
4 6
2 2/e
\°s k' 1.2 3.4 5.6 )
+2 4 6 8 * V
2 2 2 2
k ' i-2 34 5-6 7-8^
"\ /' Y lx' ' ' "
W
2
2
2 4
2
4-JL_J__jA
+ I>;Wiosg ^' 2
2 i-2
4 6 V 34 5-6/
"t~ 9
2 9
2 2 /E g ^'
2 4 6 8 V i-2 34 5-6 7.8^
+
The equations (3) are very convergent for /£'<o.i, (£^0.995), and satisfactory accuracy will be attained down to £ = 0.985, thus covering the range of values for which interpolation in Legendre's tables becomes difficult. For values of k near 0.985 it is perhaps more accurate to calculate M from elliptic integrals F and F with a modulus k greater than k. The modulus k/ which is complementary to k is smaller than k\ and the values of F and F calculated from the series formulas (3) putting k in place of k' converge more rapidly than the values of '
i^and iTwhen calculated by the same series formulas. The formula for making the transformation is not quite so simple as (2). It is most conveniently written J
^oveA Formulas for Mutual and Self-Inductance
1
it'-
l+k (l+kf [4]
Wlien the distance between the circles is large, formula (1) becomes unsuitable for calculation for two reasons, {a) because 7 falls outside the range of Table XIII and {b) because the quantity
( - — k )F — -fE comes out as the small difference of two large
quantities. The use of formula (4) overcomes the first objection, but makes the matter still worse as far as the second is concerned. We may, however, express (1) in terms of a series by means of the 3 well kiiown expressions of Wallis
fjS + 1 }"3J •<*""> r **
1
2-4-6 2ft
n
2[_ t {
- - - •
J JJ
B *_i 2'4'6
.3-5----(~-D|;*» 1
2^
2L (2/2-1)
1 [
Substituting these values in (1) we find
^VT.[x + Jr +
the general term in the brackets being ^^ + + ,...] [5 ]
( 3-5-7' - -(2tt+ i) V (2» + 2) =[ 3-5-7- -(^ + i)T ^ \4-6-8- • -(2^ + 2)7(2^-1) 'L_4-6-8 • • • 2n J(2n — i)(2n + 2)
For values of k up to 0.1 (7 = 5? 7) the series (5) is very convergent, and may be used for values of k up to 0.2 (7= n?5) without serious labor. In the latter case and for still larger values of k> we may calculate M in terms of the smaller modulus kx of formula (2). This last expression becomes on expansion 3 Greenhill's "Elliptic Functions," pp. 9, 176. —
io Bulletin of the Bureau of Standards _voi.8,No.z
ir-^Vs[i+§v+gV+^«+ • • •
•] [6]
the general term in the brackets being
(n±i_ 3'5'7 '
Tyn
' ' ' (g» + i)
\[
\2n + iJ[_4.6'8 .... (2^ + 2)J
The series (6) converges more rapidly than (5), and may be used
with ease for values of kx as great as -, (71 =i4?5), which corre- 4 sponds to £ = 0.8, (7= 53? 2). To recapitulate For values of k between zero and 0.2 use (5). (1)
(2) For values of k a little larger and up to 0.8 use (6).
(3) For values of k between about 0.7 and 0.985 the elliptic integrals in (1) may be conveniently taken by interpolation from Legendre's tables or from Table XIII. (4) For values of k greater than about 0.7 we may use (4).
(5) For values of k greater than about 0.985 we may use (3). It will be thus seen that the formulas overlap, so that it will be
possible in every case to calculate the mutual inductance by at least two different formulas, the less accurate serving as a check on the more accurate. The choice of formulas is considered more in detail on page 19. WEINSTEIN'S FORMULA
Weinstein * gives an expression for the mutual inductance of two coaxial circles, in terms of the complementary modulus k f used in the preceding series (3). Substituting in equation (1) the values of F and E given above we have Weinstein's equation, which is as follows:
<>+&•+&•+$&+ )} M
4
Wied. Ann., 21, p. 344; 1884.
1
Grfver] Formulas for Mutual and Self-Inductance 1
This expression is rapidly convergent when k' is small, and hence will givean accurate value of M when the circles are near each other. Otherwise formula (i) may be more suitable. NAGAOKA'S FORMULAS 5 Nagaoka has given formulas for the calculation of the mutual inductance of coaxial circles, without the use of tables of elliptic integrals. These formulas make use of Jacobi's ^-series, which is very rapidly convergent. The first is to be used when the circles are not near each other, the second when they are near each other. Either may be employed for a considerable range of distances between the extremes, although the first is more convenient. The first for- mula is as follows: M= 1671*^Aa~ ? (i+ f €)
= 47T 4Aa{\irq f (1 + e) } [8]
where A and a are the radii of the two circles. The correction term € can be neglected when the circles are quite far apart.
H0""<0
s-
1 -!* ¥
_r _Tl(A-a)* + d*
%
i + V# r i -yl(A + af + d*
d being the distance between the centers of the circles, and k' the complementary modulus occurring in equations (3) and (7). Nagaoka's second formula is as follows :
M= 4 7rV^ 2(l ^ 2 [1 + 8ft(i - ft + 4?x 2 )] log - 4) [9] ^ )8 j I = 4W^- _2 [1 + 8ft - 8ft + ej log i— 4 2
2(I ^ 8
|
*)+
j
_i — ^k 2^JAa
.
M
1 + 4~k ^{A+af + d
3 4 5
ei = 32?i -4o?i + 48?i
6 Phil. Mag., 6, p. 19; 1903. Recently a third expression has been found by Nagaoka (Tokyo Math. Phys. Soc, 6, p. 10; 1911). (See p. 187 below.) 12 Bulletin of the Bureau of Standards \voi.8,No.i
k is the modulus of equation (i), but is employed here to obtain the value of the ^-series instead of the values of the elliptic integrals employed in (i). This formula is ordinarily simpler in use than it appears, because some of the terms in the expressions above are usually negligible. For a third formula see page 187. Nagaoka has recently published 6 tables which materially reduce the labor of calculation with these formulas. These are reproduced as Tables XV and XVI of the appendix. From Table XV we ob-
tain directly the small difference q or qx — with q - or qx as
argument. The same table gives also the corresponding values of e and log 10 (1 + e) for use in the formula (8). To calculate q or q we enter the table with - or — as argument. 1 22 II The difference corresponding in the table when added to - or -i- gives the value of q or q x to a first approximation. This will be suf- ficient except for the larger values of q or q x which are tabulated here. For these it is sometimes necessary to use this first approxi- mation as argument to obtain a more accurate value of q or q x .
Table XVI gives the values of €x and — e/ for different values of qx and is useful in calculations with formula (9).
For circles at some distance from one another q becomes small, and the expression for / given above becomes inconvenient, because k* is so nearly equal to unity. In this case we may calculate /from the somewhat more complicated expression
/-
'
*
(i + £')(i + a/£') 2
the values of k and k' being calculated from the formulas already given. The same applies to the calculation of l in formula (ga\ x
when the circles are very near together, and consequently q is very x
small. For this case we use the expression 2 k' h (i+£)(i + V£) s
6
Jour, of Coll. of Sci. Tokyo, vol. 27, art. 6; 1909.
d
crTver] Formulas for Muttial and Self-Inductance 13
MAXWELL'S SERIES FORMULA
7
Maxwell obtained an expression for the mutual inductance be- tween two coaxial circles in the form of a converging series which is often more convenient to use than the elliptical integral formula,
and when the circles are nearly of the same radii and relatively near
each other the value given is generally sufficiently exact. In the
following formula a is the smaller of the two radii, c is their differ-
ence, A — a, d is the distance apart of the circles as before, and
r= -y/c + d
2
The mutual inductance is then
M= AnraX
^
fi
log
r\
1
c
2a
W
+ 3 d f + 3 cd>
—
16a
( -\
^ 2
2
c 3c* — 2
c
s
— 6cd
-0 2a 16a 2 48a 3
When c and d are small compared with #, we have for an approximate value of the mutual inductance the following simple 8 expression:
M=4ira\ log — -2\ [11]
When the two radii are equal, as is often the case in practice, the equation (10) is somewhat simplified, as follows: d V 2
M= \iva\ log (-£)-( + 1 6a 2 2 ) [12] The above formulas (10) and (12) are suffi- ciently exact for very many cases, the terms omitted in the series being unimportant
when — and- are small. For example, r if Fig. 2 a a
million. If, however, d= a, this term will be more than one per cent, and the formula will be quite inexact. 7 Electricity and Magnetism, Vol. II, $ 705. 8 This is equivalent to the approximate formula given by Wiedemann, M=4irallog — — 2.45 J>, where / is the circumference of the smaller circle and c is the same as r above. 21674 — 12 2 I
14 Bulletin of the Bureau of Standards [Voi.8,No.i
9
Coffinhas extended Maxwell's formula (12) for two equal circles
by computing three additional terms for each part of the expression. This enables the mutual inductance to be computed with consider- able exactness up to d=a. Formula (1) is exact, as stated above, for all distances, and either it or (8) should be used in preference to
(13) when d is large. Coffin's formula is as follows:
& d\ 16a 2 8x128a 4
128^ 2x128^ /
8
_/ d^ 247^ 7795^
+ 6xi28V _ 8x128V
\1 -.
+ i6a 31J* ,
r
V 2
i6xi28^ J]
L 3J
We have extended Maxwell's formula (10) for unequal circles as 10 follows:
2
c + 3d
2 s
+ $cd 2
+ 42c d — 1 5a 2 2 74
M= 47m log — 2a c 16a 2 c 32a 3 1 7 c*
1024a 4
3c -d 2 — 6cd
2 4 2
i9<:
5
+ 30<?V -45a/ ^ \ ,( 2 ~
c
~7
c
z 2
2048a 5
)"(
/ \
~T~
2a T-^a"
2 I
48a 3
5
,
£9g" + 534^ - 93^* 379^ + 303°^ ~ 1845^ r
I4 --i
4 ] L
6144a 61440a 5 /}
1
Nagaoka 11 has confirmed this extension by expanding for- mula (9). He carried out the expansion, however, no further than terms in — and d' a When <r=o, this gives the first part of series (13). When ^=0, the case of two circles in the same plane, with radii a and a + c, we have
M= Aired log
^ s
(1 8a/
—^\
+—
c
+ 6a"
2a -z-j
( 1
1
c%
i
c
32a
*
^—1 - V" +
7
+ 1024a
3
2048a
—1
{A
4 — T 9^
5
5 • • •
\
)
/
-/"
V
2+ /--Jf!
2a i6a
L + _i?- +
+ 6144a
_ 379^
61440a ^
2
48a 3
^ 4
5
5
,
V
/J
r
L isl
DJ
9
J.G. Coffin, this Bulletin, 2, p. 113; 1906.
10
This Bulletin, 2, p. 364; 1906.
11
Iyoc. cit., p. 11.
Grower] Formulas for Mutual and Self-Inductance 15
These formulas (14) and (15) give the mutual inductance with
great precision when the circles are not too far apart. The degree of convergence, of course, indicates approximately in any case the accuracy of the result.
HAVELOCK'S FORMULA
In 1908 Havelock 12 published a paper in which the calculation of mutual and self-inductance is made to depend on the evalua- tion of certain definite integrals of Bessel functions of the form n ^ViCaO/iC^aO f1 ^fJL 'These he expands in the form of series, l '
which fall into two classes, those suitable for small values of ft, and those suitable for large values of ft. In the case of the latter, he gives the expressions for the general terms of the series, so that these may be extended as far as desired. In the case of the former only a few terms are given, and the derivation of further terms is very tedious. He considers first the mutual inductance of two coaxial circles, and points out that the solution may be made to depend on either of two of his integrals. He does not, however, write out the for- mulas. It is a simple matter to carry out the necessary substitu- tions, and we find for circles near one another
2+ +
~V i6V^/~2048V^j /J
This expression bears some resemblance to Maxwell's series for- mula (10); it is, however, simpler for use in calculation. To obtain the coefficients of further terms by Havelock 's process would require a good deal of labor. We notice, however, that, putting A = a, the formula becomes the same as Coffin's formula (13) for equal circles. We may evidently, therefore, use the coefficients of the higher order terms in Coffin's formula to obtain an extension of the above, and find the expression 12 Phil. Mag., 15, p. 332; 1908. 16 Bulletin of the Bureau of Standards [Voi.s, no.i
A
16 1024
2 3 K
|L 128 2X128 J r
^hV^ 7^.J^+
V 16 2048
+
6X128
2
8 x128
s
•
-|
)
[16]
where r2 = c + d2 i a = -- Aa The expression thus extended 13 gives very accurate results for values of d almost as great as the radius a. For a given degree of convergence it requires only half as many terms to be calculated as does formula (14), and is much easier to calculate. The second formula derived from Havelock's paper is not so gen- erally useful, being rapidly convergent only for values of d greater than about $A. It is
M= VA 1 3 / 1+ a 2 \A 2 15/ A )Y2+ i{ 1 + ^A + 2 a2 2 a\A A^ {
]
d«
[17]
^iS/ a2 a 4,
a6 a 8 \A*
+
y( i+ioz2+2 °^ +io^ + z«; d*
693/ a2 a* a6 a8 a^XA^ "I
A )d 10+ "j
10
For the case of d= 10 A and a as great as A, only three terms have to be calculated to obtain M to about one part in a million,
and for a smaller value of -^ the convergence would be more rapid still.
MATHY'S FORMULA
In an interesting paper in the Journal de Physique for 1901, 1 * B.
Mathy obtained a formula for the mutual inductance of two circles, 13 Mr. T. J. Bromwich, of Cambridge, England, has recently communicated to us the same formula, without giving the proof, including however terms no higher than those in a 3 .
14 Jour, de Phys., 10, p. 33; 1901. G°roveA Formulas for Mutual and Self-Inductance IJ
in which the elliptic integral of the third kind, on which the mutual inductance depends, is expanded in a manner still different from that adopted in any of the preceding cases. It is expressed in terms
of hypergeometric series involving the absolute invariant J of the Weierstrassian p function. The final expression as found by Mathy is incorrect as regards the coefficients of the hypergeometric series. 15 The corrected expression, using the notation of this paper, is as follows :
K ** p J i/-^ 1
Q 17^
f
4^=L(^+i2^v)4^i^^Vi2
f
,
5
i2' 2 ' / ; 6-j/ 3 V^
where x* = a 2 +A +d 2 2
P= 1.311028777
2 = 0.599070117
and [+^)7 v 1-7 1-3.7(7+1)
^+l)(^ + 2)-/3(/3+l)(/3+2^ 3
1.2.3.7(7+1X7+2)
This formula is by no means so formidable to use as might be expected, since the constants which enter and the coefficients in the hypergeometric series may be calculated once for all. Using seven place logarithms we find
15
Grover, this Bulletin, 6, p. 489; 1910.
8
1 Bulletin of the Bureau of Standards \yoi.8,No. i
P —P=, = 8.9816199
lo&iolT— =97597712 log10
!ogio -J^= 9-6581974 lo ^o -Mr = 8.8800461
The coefficients # a # 1} 2, 3
in each of the four series are given in
Table XVII. For practical purposes the formula should be used
only for values of «jJ-lLE smaller than about 0.2.
For the special case -J
r
= o, it is of interest to note that the
mutual inductance is given by the simple expression
M_
47T
which, remembering that x? = $6A 2 a 2 in this case, becomes
M= 47r-yjAa(P- 0) = 2 4?r(o. 1 12888542) ^Aa
= 1.418599262^/^^ [19]
If we introduce the distances r and r (Fig. 1) into the formula
x 2
for -W ^ , we see that the necessary and sufficient condition that this remarkably simple formula 15a may be used is that r* = 2r 2 2 , or
k' — k = —7=. That is, the greatest distance between the two cir-
cumference must be ^2 times the shortest distance between them. The most important cases satisfying this condition are
a
d
A
1 2A Equal circles.
3-2-V2 Circles in the same plane.
1
2 5V7 A
15a Nagaoka has recently shown (Tokyo Math. Phys. Soc., 6, p. 10; 191 1) that formula (19) may be derived from Maxwell's formula (1). Rosa Groverj ~ Formulas for Mutual and Self-Inductance x9
The convergence of the formula (18) will of course be satisfactory d for moderate deviations on the either side of the ideal ratio of A" but the formula must be regarded as of more limited application than most of those above. It gives, however, a very rapid and accurate means of checking other formulas, since in the ideal case the mutual inductance can be calculated by (jo,) to any number of decimal places desired, according to the number of figures retained in Stirling's constants P and Q. CHOICE OF FORMULAS With so many to choose among, it is possible to select a favorable formula for any individual case. For this purpose r and r2 the 3 ,
longest and shortest distances between the circles, need to be con- sidered, since on their relative values the convergence or convenience of the various formulas for calculation depends. The following r table gives roughly the range of values of the ratio -? within which ri the different formulas are capable of giving the best results. Since, however, the determination of such limits is somewhat arbitrary, the values given here should not be regarded as more than a guide. In the case of those formulas which occur in the form of a series
the limiting value of the ratio — has been calculated which makes the last term included not greater than one ten-thousandth of the r whole. The values of ~-L for Nagaoka's formulas have been calcu- li lated for the limits of his correction tables.
SUMMARY OF FORMULAS FOR CIRCLES
Most favorable
Range of values
Formula ^ values of —^ for
of
*i equal circles
Weinstein's (7) O to 0.25 O to 0.5 Maxwell's (10) O to 0.02 O to 0.04 u (12) O to 0.14 O to 0.3 u O (14) to 0.22 O to 0.45 (< (3) O to 0.2 O to 0.4 << (2) 0.02 to 0.20 O.04 to 0.4 20 Bulletin of the Bureau of Standards [Vol.8, No. I
Range of values Most favorable
d
Formula
< values of -3 for
equal circles
Havelock's (16) to 0.4 o to 0.9 Coffin's (13) o to 0.9 Nagaoka's (9) 0.04 to 0.4 0.08 to 0.9 Maxwell's (4) to 0.75 o to 2.25 C< (1) 0.2 to 0.7 O.4 to 2 Mathy (18) 0.65 to 075 1.75 to 2.25 K (19) 1
Nagaoka's (8) 0.3 to I greater than 0.6 Maxwell's (6) 0.6 to I « 1.5 Havelock's " " (17) 0.9 to I 4 Maxwell's o.q8 to I cc u IQ (5)
EXAMPLES TO ILLUSTRATE AND TEST THE FORMULAS
EXAMPLE 1. MAXWELL'S FORMULA (1). FOR ANY COAXIAL CIRCLES d~20 L,et a = A = 25 cm, Fig. 3, </= 20 cm.
a=25 50 £= = 0.9284766 = sin 7 V2500 + 400 7 = 68° n' 54 88 = 68?i98578. r/
From Legendre's tables, we obtain
log F= 0.3852 1 9 1
/^_£V--^=
FiS- 3 log 2?= 0.0547850 \k )
M *
0.5318500
k
4^ = 100 .*. M= 167.08562 cm.
To facilitate calculations in such problems as this, we have pre-
pared Table II, which gives F and log F E and log F, as functions y
of tan ' 7. In the above case tan 7 = 5- = 2. °5, and from Table II we 20 can take the values of log F and log E directly, avoiding the calcu- lation of 7 and the interpolation for log F and log E in Legendre's tables (or Table XIII). This is only applicable for circles of equal radii, and is especially advantageous when tan 7 is one of the values given in the table, when interpolation is unnecessary. 2 :
%°rover
Formulas for Mutual and. Self-Inductance 21
The above problem may also be calculated by means of Table I,
taken from Maxwell, as follows
logio M for 68 ? 1 = 1 7 2 30634 .
\ira
for 68? = 1.7258281
for 68?io8578 = i.7257888 = log —M
\ira
.\ M= 167.08546 cm, agreeing almost exactly with the above value. The calculation of mutual inductance by the above methods is simplest for circles not near each other, as then the values of log F,
log E, and log Mr=p are very exact when taken by simple inter- polation. When 7 is nearly 90 °, however, second and third differ- ences have to be used in interpolation.
EXAMPLE 2. MAXWELL'S SECOND EXPRESSION (2). FOR CIRCLES NEAR EACH OTHER Let a = A = 2$ cm, d= 1 cm
In this case k= sin 7= ,- = 0.9998002 7=88° 51' 14"
This value of 7 is so nearly 90 ° that it is difficult to obtain accu-
rate values of F and E from tables of elliptic integrals, or of from Maxwell's table. We may therefore use formula (2) instead of (1). rx = V2501 = 5 ao1 nearly, r.z = 1.0 .-. k,= sin 7^^-^ = 0.9607920 7i = 73° 54-' 9- //6 7 = 73 -902687 From Legendre's tableslfor y1 = 73^902687, F = 2.7024545 1
or Table XIII, E = 1.0852170
x
^-^ = 1.6172375
8ttV^ 2 QQ7r STr^/Aa/^ \ '
22 Bulletin of the Bureau of Standards [Voi.8,No.i
EXAMPLE 3. FORMULA (3). SERIES FOR F AND E, CIRCLES NEAR EACH OTHER Suppose that, in the last example, we calculate F and E by means of formula (3), instead of taking them from Table XIII.
A = a=25, d=i.
2501 2501
.*. ^=5.2989471 is = I.OOO9594
If these values of .Fand E be substituted in formula (1), k being
0.9998002, we obtain M= 1036.6652, which is very closely the same
value as by formula (2).
EXAMPLE 4. FORMULA (3). SECOND CASE, CIRCLES NOT NEAR
^ = 25, # = 20, d= 10 cm. (See Fig. 1.)
_ 4X20X25 _l6 .£/2 = jt_
17 "
k% 2 2
(45) + (io) 17
log|7 = -log (i6xi7) = ^ lo g* 272 =2.8029010
2
= •° 26 5 I 3a
4 (
log
F~ r)
9 log _ = 00 ° 7962
64 f( l 6)
g = •°00° 312
256V~**
256 k> go)
90/
^(^4-1.27) = -0000014
.-. F = 2.8302430
I+^(l0g|7-j) =1.0677324
1 - 20 = • 000 ° 381
ifirf^F- )
^(iogJ-1.25) = -00000x7
.-. .£• =1.0688878
cooler] Formulas for Mutual and Self-Inductance 23
To find the value of Mwe now use equation (1). 2 &-*>- £[ = 0.885388 Multiplying by 477^^ = 477^500 gives
M= 248.7875 cm.
EXAMPLE 5. FORMULA (4). CIRCLES NEAR TOGETHER
A = a = 2$ d=\
2^Aa 50 ,
,
k = .
v
x
— =
-
,
=0.9968154
*"- - ^5=0.0063593015
(^ +«)+*
£/2
^' = = OGI 5949004
(l+k)* *
l0g e ^7 = l0g, 2507.9937 = 7.8272373
^-Ylop- 4 - A - °- QOOO°43 _ Jp
~
4 V
ge
V V 7T82727i6-
°
1 + ~- (l°ge ^-5) = I -ooooo 93 = £"
2 \ "" #
'
= 3-9323856
£(l+£)
E °-(i + £) = 2.0032134
k
1.9291723
Multiplying by ^ir^Aa gives
Af = 606.0674 cm.
If we calculate M by formula (3) we find that, to obtain the same precision, terms in k n in the series for i^and E have to be included, and we find ^=606.0678 cm. 24 Bulletin of the Bureau of Standards [Voi.8, m.i
EXAMPLE 6. FORMULA (5). CIRCLES FAR APART
A = a=io d=ioo
20 , i
„
k= i
- = -===0.19611615
V10400 V26
1+h 2
=1.02884616
4
4
y^£ =0.00086684
—245
— kr =0.00002723
5 12
6615
2
£8 = 0.00000088
128
Sum =1.02974111
log sum =0.0127281
log k* =3- 8 7754oo
2
J~Aa
7r
.
log —7— = 1.3922398
4
log Af =1.2825079
J/ =0.19164962 cm.
If formula (1) be used, and the values of and be taken by F E
interpolation from Table XII, the value ^=0.191643 is found, which is in error by more than 5 parts in 10000. Using the formula (6) terms in k* only need be calculated, and we find o. 191 64958, M= which differs by only one part in five million from the value given by (5).
EXAMPLE 7. FORMULA (6). CIRCLES NOT NEAR TOGETHER
A = 25 a=2Q d=\o
^i = V3625 r2 =Vi625
Grfver] Formulas for Mutual and Self-Inductance 25
1
+ 1/^! = 1.01469245
2
log sum = 0.0064929
^kf = 0.00035978 log ^f = 2.9447972
^-K = 0.00001028 log ~JAa = 1.5456049
122 ^
k* = 0.00000032 log — = 0.4968950
Jkf
Sum= 1. 0150628
-• —= 3- * 397496 cm
By formula (8)
M
— = 3.1397486
If formula (1) is used and the elliptic integrals be taken from
Table XII by interpolation trie value M = 3.1397656 — is found, which is only five in a million in error.
EXAMPLE 8. WEINSTEIN'S FORMULA (7). FOR ANY COAXIAL CIRCLES NOT TOO FAR APART Take the same circles as in example 4. ^ = 25, a=20, £=5, d=io k f2 = —y lOg 4— I = I. 80290I Iy K
I +^ =1.0441 I76
/8
I+^-k n = 1.0004053
^k n = .OOI7842 -—^k'* = .0000245
— Lk'*= .00008^1
256
° 7 , k'*= .OOOOOI2
65536
JSQI3 k»,. 1.0004^10=6"
^°
/X r%
= .0000042
16384 1_
Sum = 1. 045991 1 = i?
i? log(-i- i) = 1.8858184; ji? log
(£- 1)- CJ = 0.8853874
Multiplying by 4^^500 gives ^=248.7873 cm, agreeing almost exactly with the value previously found, example 4. 26 Bulletin of the Bureau of Standards [Voi.8,No.i
EXAMPLE 9. NAGAOKA'S FORMULA (8). CIRCLES NOT NEAR
TOGETHER
A = a = 2$ d=20 (See Fig. 3.)
/ I — *Jk' I 0.3QOS8l7
From Table XV, q =0.00005269
2
.-.
q =0.12139519
-3- log $r = 2.6263022
2
From Table XV, log (i + e) =0.0002775
log 4007T 2
= 3.5963598
log M =2.2229395
.*. M = 167.08577 cm
or about one in a million higher than the value found for the same circles in example 1.
EXAMPLE 10. NAGAOKA'S FORMULA (8). CIRCLES FAR APART
A = a=\o d=ioo
100
k' = 7 = 0.98058073
V10400
k= 1
= 0.10611615
y 10400
i-f^/F= 1.9902427
ri + ^+v^r ao° 24512756
(i
The differences ^ — - and e are negligible, so that we have
4
M= 16 ir
2
^Aa ( - Y = 0.1 91 64966 cm
g
G?wer
*
Formulas for Mutual and Self-Inductance 27
which is in very close agreement with the valves found by formulas (5) and (6) and Havelock's formula (17) for the same pair of circles.
If we calculate - by ' the formula - ; ^7= instead we find diffi- 2 2(i + V^) culty in obtaining (1 — ^//F) with sufficient precision. The value of M found by using this formula for — and with seven place logarithms is in this case ^=0.19164980, or about one part in a million differ- ent.
EXAMPLE 11. NAGAOKA'S SECOND FORMULA (9). FOR CIRCLES NEAR EACH OTHER A = a = 2$ d=4 50 ,_ k = sin 7 = /— = 0.99681535 ^k = 0.99840640
,, J(A — a)7 + d
k n2 = V
2
= ^4 = =
r
2
^A + af + d2 V2516 0.006 6x £$9 63014*
,
^
l b>*
2 ( 1+ k)(l + ^ ^ = 0.00039872542 = ^
as ( - and higher powers can be neglected. J
From Table XVI l0ge W = 1 °g 2 5°7-99 I 9 = 7-8272376 '
- e/ = 0.000001 28
%i + € = 0.00318852
i
[i+8ft + e/] log, Qj- 4 =3.8521929 =P
jjj^ji- o.50079850=G
.*. M=\Tr^~Aa • P<2 = 606.0674 cm
which is exactly the same value as was found for the same circles in example 5. 28 Bulletin of the Bureau of Standards [Vol. 8, jvo. i
Using Table II for the above problem, where tan 7=12.5, we have log ^=0.5932708 and log ^=0.0047004. Using these values in formula (1) we obtain for the mutual inductance M = 606.0666 cm which differs from the value by Nagaoka's formula by 1 part in a million.
EXAMPLE 12. MAXWELL'S SERIES FORMULA (10). FOR ANY TWO COAXIAL CIRCLES NEAR EACH OTHER ^4 = 26 ^ = 25 d=i c=i r=-yj2
= log, — = 4.9517438
Sa 200
Since r= A 2,
1-
log, —
;
..
A 2
C C
l-\ = 1.0200000 2+—
,
= 2.0200000
2a 2a
2
-
— °%f, d = —
2 2
c + 3d
, , = .0004000 .0002000
i6# 2
i6^ 2
c
3
+ \cd? o
+
^
3
- 6V^ = —2
/-
3
.0000080 ,
^3— .0000067
32a „
^ 48^
^
1.0203920 = 2.0197933=0
B\og~-= 5.0527192
C = 2.0197933
2?log—-— Cl= 3.0329259 Multiply by 47777= ioo7r and
^=952.8218 cm.
This formula would be less accurate for the circles of problem 4, but is accurate for circles close together, as this problem shows. EXAMPLE 13. MAXWELL'S FORMULA (12). FOR CIRCLES OF EQUAL RADII NEAR EACH OTHER A = a= 25 d= 1 -^=200 log, 200=5.298317
Sa/ vf \
loge "7\ I+
rS? / =I 000 3
v y
- °X5-2983 I 7 = 5-29990
000 10
2+
( £)=i.,29980
Multiply by 4.77a = ioott
M= 1036.663 cm
nearly agreeing with the more exact value found under problem 2. Grower] Formulas for Mutual and Self-Inductance 29
This is a very simple and convenient formula for equal circles and gives approximate results for circles still farther apart than in this problem.
EXAMPLE 14. HAVELOCK'S FORMULA (16). FOR CIRCLES AT MODERATE DISTANCES ^ = 25 a = 20 d= 10 .'. *=5 c
r 2
1 SJAa
— =
r=5S5 a = —r~=- -
Aa 4 r
log, 16= 2.7725887
l+-±La= 1.0468750
16
^-oT= —O.OOO9155
=z°?== O.OOOO334
j<2 = —O.OOOOOI5
2.128
Sum = 1.0459914
Multiplied by log, 16= 2.9001037 = i?
2+—pa= 2.0156250
^^=-0.0009461
i
a6 = O.OOOO393
6.128
779S
J 4.
= — O.OOOOOIi
»oc
3
8.128
Sum = 2.0147164=6"
B-C= 0.8853873
Multiplied by ^ir^~Aa = 248.7873 cm = M
which agrees exactly with the value found in example 8. 21674 — 12 3 3<D Bulletin of the Bureau of Standards \voi.8,No.i
If the example 1 2 be calculated by this formula, no terms of order
higher than a 2 need be calculated, and
M= 952.8221 cm
Formula (10) ^=952.8218
Formula (3) ^=952.8219
EXAMPLE 15. COFFIN'S FORMULA (13). EXTENSION OF FORMULA (12) FOR CIRCLES OF EQUAL RADII A = a = 2§ d=i6 -^=12.5 log, 12.5=2.5257286
First series of terms = B= 1.074478
Second series of terms = C= 2.023220
,'AB log ~r— = 0.690620
47m = ioo7r .\M= 216.9647 cm.
This agrees with the value given by formula (1) within 1 part in 200,000. As the distance apart of the circles increases the accuracy by this formula of course gradually decreases. EXAMPLE 16. FORMULA (14). EXTENSION OF MAXWELL'S FORMULA (10) FOR CIRCLES OF UNEQUAL RADII A = 2$ a = 20 £=5 d=io r=V?T^ = 5V5 2 log, y = log -^=2.6610169 e
Q
First series of terms = B log, — = 3.112060
Second series of terms = C =2.122114
0.989946
multiplying by 477-0 = 8o7r M= 248.8006 cm.
This result is correct to 1 part in 19,000 (see examples 4, 8, and 14). Using only the first three terms for B and C (that is, formula 10), the result would be too large by 1 part in 1750.
EXAMPLE 17. HAVELOCK'S FORMULA (17). CIRCLES FAR APART
a=\o = A d=ioo
a
-r=I
A
-^ = 0.1
A d
Grover] Formulas for Mutual and Self-Inductance 31
i-3. 0.9700000
2 d"
15.
8 «*- 0.0009375
35.
16
»$-- 0.0000306
3*5.
128 »4- 0.00000 10
'Sum = 0.9709079
Multiplied by —27r
2
-^
A—
a
=
2 2
o. 191 64958 cm = M
which is in exact agreement with the value found by formula (6).
EXAMPLE 18. MATHY'S FORMULA (18)
A = 25 <2=20 ^=40
X = 625 + 400 + 1600 = 2625
2
Aa _ 500 _ 4
l?~5625""2i
^r* + i2 A = 9890625 V
i log (#* + i2^4V) = 1.7488059
4
log =I 6703234
[(F+^?vyi]
-
2 2
i-3 6 ^4
-^=-0.3061225
<2
=-0.l780II3I
•%/ /^
log# = 2.5008952
Using the constants in Table XVII we calculate the four series
F(±, JL l, 2
, jSJL, B, 3,,) F(-±, L, 1, g ) F (±, 13, 3,
\I2 12 2 ) \I2 12 2
/ 12 12 2 /\I2 12 2 \ / /
1.0022006 I.OII2962 O.9969192 I.OO95357
O.OOOO357 0.0002173 —O.OOOO473 O.OOOI784
0.0000008 0.0000049 — 0.0000010 0.0000040
I.002237I I.OII5184 O.9968709 I.OO97181
32 Bulletin of the Bureau of Standards [ Vol *, No. 1
From these, using the values of the constants already calculated, we find the four terms in the formula for M C= 26.981438 G= 25.447327 B= — 0.639427 H= — 0.966215 C-D= 27.620865 G-H= 24.481112 M = 27.620865 - 24.481112 = 3-139753 — By Nagaoka's formula we (8) M = 3.1397496 — find 47T
(6) «
M
— = 3.1397486
«
4 7r
Mathy's formula suffers here under the inconvenience that is M given as the difference of two quantities considerably larger than itself.
EXAMPLE 19. FORMULA (19). FOR CIRCLES SATISFYING THE CONDI- TION t?=2i? OR k'=k=-^ V2 A = a = 2$ ^=50 ^=1.41859262 ^A~a• •
= 35.4649816 .... cm.
By Nagaoka's formula (8), M= 35.464975
« (6), ^=35.464981
« (1), ^=35.46481
We see that the formulas (8) and (6) here give an accuracy limited
only by that of the logarithm tables. The result found by formula (1), using Table XIII, is, however, affected by the fact that the value of the quantity in the parentheses (1. 4239167 — 1.3110287), is only about a tenth as large as the numbers of which it is the difference. Groler] Formulas for Mutual and Self-Inductance 33
2. MUTUAL INDUCTANCE OF TWO COAXIAL COILS
ROWLAND'S FORMULA
Let there be two coaxial coils of mean radii and a, axial breadth, A of coils bx and b21 radial depth ct and c2) and distance apart of their mean planes d. Suppose them uni- formly wound with n x and n z turns of wire. The mutual inductance of M the two central turns of the coils (Fig. 4), will be given by formula (1) or (7), or any one of the foregoing formulas for the mutual inductance of coaxial circles adapted to the particular case may be used, and the mutual induc- tance M of the two coils of n x and n 2 turns will then be, to a first approxi- a.. mation, M= n n M^ x 2
The following second approximation was obtained by Rowland by means of Taylor's theorem, following Maxwell, § 700: dM 24 w « b?) n.11 dx" tPAf. dV^) + • da" + c dA % ]
If the two coils are of equal radii but
Fig.
unequal section, M ~".+i$V+VF&+V+<rfg} dM n [20] n,n
If the two coils are of equal radii and equal section, this becomes
n,n 9 ° 12 dx 2 da 2
34 Bulletin of the Bureau of Standards Woi. 8, No. i
The correction terms will be calculated by means of the following:
3?-$-'>'-(-'£S>} r ,
dx% a[ i —k 2
The equation (21) is equivalent to Rowland's equation, where 2? and 277 are the breadth and depth of the section of the coil, instead of b and c, except that there is an error in the formula as printed in Rowland's 16 paper, f and rj being interchanged. The equations (22) are equivalent to those given by Rowland, being somewhat simpler. 17 Formula (21) gives a very exact value for the mutual inductance of two coils, provided the cross sections are relatively small and the distance apart d is not too small. But when b or c is large or d is small the fourth differential coefficients which have been neglected become appreciable and the expression may not be sufficiently exact. RAYLEIGH'S FORMULA
Maxwell 18 gives a formula, suggested by Rayleigh, for the mutual inductance of two coils, which has a very different form from Row- land's, but is nearly equivalent to it when the coils are not near each other. It has been used by Rayleigh in calculating the mutual inductance of a Lorenz apparatus and by Glazebrook (Phil. Trans., 1883) in calculating the mutual inductance of parallel coils of rectangular section employed in a determination of the ohm. It may also be employed in calculating the attraction between two 19 coils. It is sometimes called the formula of quadratures, and is as 20 follows:
M=^M + M + M + M + M + M + M + M -2M^\ [23]
1 2 3 l 6 t 1 i
16
Collected Papers, p. 162. Am. Jour. Sci. [3], XV, 1878.
17
Gray, Absolute Measurements, Vol. II, Part II, p. 322.
18
Electricity and Magnetism, Vol. II, Appendix II, Chapter XIV.
19
Gray, Absolute Measurements, Vol. II, Part II, p. 403.
20
This Bulletin, 2, p. 370-372; 1906.
G°rover
Formulas for Mutual and Self-Inductance 35
where M x is trie mutual inductance of the circle 2 and a circle through the point 1 of radius A — and c 2 -, similarly for the others,
Fig- 5-
s
!
I
U ^ CM
i
1
l
1
I
1
^-~
r- -^
r^^^.^
7
P <!
%xT~--
°2 6
2
V
I
< &
\
Fig. 5
For two coils of equal radii and equal section this becomes
M 7"- + M + M + M -M
2 3 i Q
•)
[24]
Equation (23) is Rayleigh's formula, or the formula of quadratures. Instead of computing the correction to M by means of the differen- tial coefficients (20), eight additional values are computed, corre-
sponding to the mutual inductances of the single turns at the eight numbered points indicated in Fig. 5, each with reference to the central turn of the other coil. These M^s may be computed by any of the formulas for the mutual inductance of coaxial circles which may be best adapted to the particular case, and the values of the constants for the case of two coils of equal radii are given in the following table, the radius being a in every case. 36 Bulletin of the Bureau of Standards [Voi.8,No.i
Axial distance Radial distance r
Using (10) 2 aKt' u d +? Y 4
a d ^2
/// 2 J_ 2
2
u / **
d +
+f V 4
Using (12) d--4/2 O u d+bjl O U d+b J2 2 O u d--h\z O
MAGNITUDE OF THE ERRORS IN ROWLAND'S AND RAYLEIGH'S FORMULAS
The error e in equation (24), for two coils of equal radii a dis-
x )
tance between centers being d and section bxc (Fig. 6), depends y
on the dimensions of the coil in a manner shown by the following .21 expression
e1 oc 477-tf
3#* + 3^ — 2o£V
[25]
4800?
For a square coil the correc-
tion is a negative quantity,
showing that M by equation
(24) is too large, and the error
is proportional to the fourth
power of -3, the reciprocal of the
Fig. 6
distance between the mean planes of the coils. For a rectangular coil in which b is greater than c the correction is negative so long as b is not more than 2.5 times c. When b is still larger with respect to c the correction becomes plus, the value of M by (24) being too small.
21
This Bulletin, 2, p. 373; 1906.
— —
Grwer] Formulas for Mutual and Self-Inductance 37
Thus, for a coil of cross section 4 sq. cm, we get the following values of the numerator of (25) as we vary the shape of cross section keeping be =4. Dimensions of coil Error pre >p< Drtional to b=2 C=2 - 224 b = 2.5 C= 1.6 - 183 b= 3 c =1 -33 - 67.5 £=4 C=l + 45 1 b= 8 c = o.$ + II ,988 Thus we see that the value of M as given by the formula of quad- ratures may be too large or too small according to the shape of the section, and that the error is proportional directly to the fourth power of the dimensions of the section and inversely to the fourth power of the distance between the mean planes of the coils. When the section is small and d large the error will become negligible. The error by Rowland's formula is 22
€
* * 47rMl6c7"^44j " 4M 48c^ )
[26]
|
This is negative for a square coil, but smaller than ev For a coil of section such that b = c^2, the error is zero, and for sections such
that- > ^2, the error is positive. Thus, for a coil of cross section 4 sq. cm, we get the following values of the numerator of (26) which is proportional to the error by Rowland's formula.
Dimensions of coil Error proportional to-
b=2 C=2 64
£=2.5 C= 1.6 + 45
b=3 c =^33 + 353
£=4 c=I + h736
b=8 ^=0.5 + 3M48
Thus the error is smaller by Rowland's formula for coils having square or nearly square section, but larger for coils having rectangu- lar sections not nearly square.
22
This Bulletin, 2, p. 373; 1906.
: :
38 Bulletin of the Bureatt of Standards [Vol. 8, No. i
LYLE'S FORMULA
23
Professor L,yle has recently proposed a very convenient method
for calculating the mutual inductance of coaxial coils, which gives very accurate results for coils at some distance from each other. The mutual inductance is calculated from formula (i) or any other formula for two coaxial circles, using, however, a modified radius r instead of the mean radius a, r being given by the following equation when the section is square, b being the side of the square section
a\ i + 24a 2 [27]
< J
If the coil has a rectangular section not square, it can be replaced
by two filaments (Fig. 7) the distance apart of the filaments being called the equivalent breadth or the equivalent depth of the coil. 2 —c 2
ft
2
=b , 2 /3 is the equivalent breadth of A
[28]
2
=
2
-b 2
a , 2 3 is the equivalent depth of B
The equivalent radius of A is given by the same expression which holds for a square coil, viz
$ =a( 1+-^-!
1 V 24a 2 J
)
1
f c<
In the coil B the equivalent fila-
ments have radii r+8 and r— 8,
A ~B respectively, where
Fig 7.
The mutual inductance of two coils may now be readily calcu-
< 1 + 24a 2
/
lated. If each has a square section, it is necessary only to calculate
the mutual inductance of the two equivalent filaments. For coils mutual inductance will be the of rectangular sections, as A, B, the sum of the mutual inductances of the two filaments of A on the two Phil. Mag., 3, p. 310; 1902. Also this Bulletin, 2, pp. 374-378; 1906. —
Rosa Formulas for Mutual and Self-Inductance Grover, 39
filaments of B, counting n\z turns in each. Or, it is n x n % times the mean of the four inductances M Mu M M^ where M 13 , , 23 , 13 is the mutual inductance of filament 1 on filament 3, etc. Lyle's method is of special value in computing mutual inductances because it applies to coils of unequal as well as of equal radii,
ROSA'S FORMULA 24
Writing the mutual inductance of two coaxial coils of equal radii
and equal section as nx n % = M + 4M, where M Q Q is the mutual induct-
ance of the central circles of the two equal coils of sections bxc, Fig. 5, and J M is the correction for the section of the coil, the value
of JM is as follows:
JM=^7ra\ 6 \ -log- + -=5- + -^
[
96^ to
d 192a 2f- i2d 2 i2od*
+ d
57 60a d
2
2
504^ I024A 84 )
For a square section, when b — c^ this becomes
An/r irb\ 8a a2 b 2 3^/1 Sa 4\ ^l b% 1 r 1
The last two terms of equation (30) are relatively small, so that we may write, approximately: Anyr 2 7rb L 8a a 2 b 2} r -,
For coils of equal radii but unequal sections, the formula is,
neglecting differentials of sixth order
4M- ^m±M)±M±£h 192a: & ^_ log d ii(V + V)-3fe' + 384a 2 I
+ w+ ujp + <n [32]
960^
This Bulletin, 4, p. 348, equations (38) and (39).
4o Bulletin of the Bureau of Standards [ Vol. 8, No. I
These expressions for JM are very exact where the coils are near together or even where they are separated by a considerable distance, bnt become less exact as d is greater. They are therefore most reli- able where formulas (21), (24), and (27) are least reliable. As form- ula (31) is exact enough for most purposes, it affords a very easy method of getting the correction for equal coils of square section. Stefan's formula for the mutual inductance of two equal coaxial 25 coils (originally published without demonstration) is incorrect and is not given here. It resembles equation (29), but is seriously in error for coils at considerable distances.
THE ROSA-WEINSTEIN FORMULA
Weinstein's formula 26 for the mutual inductance of equal coaxial
coils has been revised and corrected by Rosa, and the value of JM, the correction for section, expressed separately. The expression for J Mis as follows : 26a
JM= 4™ sin 7J(,P_ E)(a + ^ + EB^ [33]
where F and E are the complete elliptic integrals to modulus sin 7, Fig. 8 (as in equation 1),
25
Wied. Annalen, 22, p. 107; 1884.
26
Wied. Annalen, 21, p. 350; 1884.
260 This Bulletin, equation (20); 1907.
4, p. 342,
\ j
Rosa "I Formulas for Mutual and Self-Inductance G rover 41
and A = T^F\ ai- a*~ a + 2a ~ 3"s) cos 7 + 8a cos* 7 * ( * 2 3
B= 2 4
lW\ ai + 7 + 2 ^a + (2^ + 3«s) cos 7 + 8^3 cos 7 J 3
The values of a,, ar and a„ are as follows: 9,
«r1 = ^_^ + For square section: o^ =
3 oa 30^
b*
Ofo = <*„ =
6o# 2 60a 2
b*
ar 9 = <*,= 72
20^ 2 200
Formula (^i,) is a very exact formula for all positions of the two coils, except when they are very close together. Weinstein's original formula, 27 which is much less accurate than (33) for coils relatively near together, is not here given.
USE OF FORMULAS FOR SELF-INDUCTANCE IN CALCULATING MUTUAL INDUCTANCE One can sometimes obtain the mutual inductance of adjacent coils, or of coils at a distance from one another, by means of a formula for the self-inductance of coils. Thus, suppose we have a coil of rectangular section, which we subdivide into three equal parts, 1, 2, 3, Fig. 9. L,et L be the self-inductance of the whole coil, L be the self-inductance of any one of the x (? & >
three equal smaller coils, and Z be the self- 2
inductance of two adjacent coils taken together. Also let M be 12 the mutual inductance of coil 1 on coil 2, or of coil 2 on coil 3, and 13 be the M mutual inductance of coil 1 on coil 3. Then,
L = 3 L + 4M + 2M 1 12 1:
Also, L = 2 A + 2^/i
2 8
M = L — zL1%
2
2
x [34]
and M = 13
Axis
Fig. 9
Wied. Annalen, 21, p. 350; i{
42 Bulletin of the Bureau of Standards [Vol.8, No. i
Formula (34) will thus enable us to find the mutual inductance of two coils of equal radii adjacent or near each other by the calcula- tion of self-inductances from such formulas as those of Weinstein (88) and Stefan (90). These latter formulas are not, however, exact enough when the section is large to permit us to apply them to coils at any considerable distance from one another.
GEOMETRIC MEAN DISTANCE FORMULA
The mutual inductance of two coaxial coils
adjacent or very near can sometimes be obtained
A r— / 4-f \B by means of the geometric mean distances. This i method is accurate only when the sections are I A very small relatively to the radius. It can often be used to advantage in testing other formulas, but not often in determining the mutual in- ductance of actual coils. Formula (10) gives the mutual inductance of two very near coaxial coils in terms of the geo- metric mean distance, if r be replaced by R, the geometric mean distance of the two sections. Formula (10) gives MQ if r be used, where r is
the distance between centers. Thus,
-J
Fig.
\
10
1 -j
^-4«(i+£)
h. log R [35]
For coils A and C (Fig. 10), R<r and J Mis positive; .# = 0.99770 r " " A " B, R>r and JM is negative; R= 1.00655 r
The same formula may also be used for squares not adjacent, but only when quite near. 28 For illustrations and tests of the above formulas, see examples 20-33, pages 44-52. 28 For other values of the geometric mean distances of squares in a plane see this Bulletin, 3, p. 1; 1907. GrTver] Formulas for Mutual and Self-Inductance 43 CHOICE OF FORMULAS
(a) For coils of equal radii and equal cross section (29) should be
used if the coils are rather near together. If the cross section is square (29) takes the more simple form (30), and in some cases this may be used in its abbreviated form (31). For coils at all dis-
tances, except near together, {^^) gives very good precision; (24) and (21) are not so accurate as this last, but give good results if the coils are far apart and their cross sections are not too large. For coils of equal radii but unequal section (32) is accurate (b)
for coils not too far away from one another. For coils farther separated (20), (23) or (28) may be used. (c) For coils of unequal radii (23), (24), (27), and (28) apply, but
unfortunately they are not as accurate as some of the others, except when the coils are relatively distant or have very small cross sections. The difficulty can be overcome by subdividing each of the two coils into two, four, or more equal parts, and taking the sum of the mutual inductances of all of the parts of one on all the parts of the other. This is a laborious operation, but in im- portant cases it should be done. As the subdivision is carried further the results will approach a final value, and hence the results themselves show when the subdivision has been carried far enough. Thus, suppose two coils A, B (Fig. 11) of square section are sub- divided into four equal parts and by the method of L,yle, formula (27), the mutual inductance of the whole of B is computed on each of the four parts of A. If the sum differs appreciably from the result
obtained by taking A and B as wholes in one calculation, then the four parts of B may be taken separately with respect to the separate parts of A. If one is doubtful whether this is sufficiently accurate,
one of the sections of A may be subdivided further and calculated with respect to one section of B, to see whether there is any appreciable 44 Bulletin of the Bureau of Standards [ Vol. 8, No. i
difference due to this further subdivision. For coils of equal radii very accurate results for near coils can be obtained much more easily by using some of the other formulas.
EXAMPLES TO ILLUSTRATE THE FORMULAS FOR THE MUTUAL
INDUCTANCE OF COILS OF RECTANGULAR SECTION
EXAMPLE 20. ROWLAND'S FORMULA (21). FOR COAXIAL COILS OF
EQUAL RADH
A = a=2$ b = c=2, cm d= io (Fig. 12.)
The mutual inductance of the two coils is
M = M + JM.
a
n n%x
We find M by formula i, 8, or 13, and 4Mby 21 and 22.
^=107.488577
50
k = sin 7 = = 0.9805807
-^2600
^ = 0.9615383
logio ^=04821754
log10 ^= 0.0207625
log ,F= 0.4821752 and log E=
By Table II, since tan 7=5,
0.0207626. These slight differences in the logarithms obtained in the two different ways amount to scarcely one part in two million of F and E, respectively, and may usually be neglected. If more accurate values are required they may be obtained by carrying the interpolations further in L,egendre's table, provided the angle 7 is obtained with sufficient accuracy. Substituting these values in formula
ic=2 ~n (
22 ) we °^ tai n
1
— 0.908177
d=lo- da 2
d 2
M = +1.063977
2
£W = 42
dx
1
1
\a=25 Substituting these values in formula
(21) we obtain
JM= .0519477
•
*.
M = M + 4M= (107.4885 + 0.0519)77
n+n
Fig. 12 = 337' 1 cm.
Rosa Crover, Formulas for Mutual and Self-Inductance 45
The correction JM thus amounts to about 1 part in 2000 of M. At a distance d=2o cm, the correction is over 1 part in 1000. For a coil of section 4x4 cm at d= 10, J M would be four times as large as the value above, or about one part in five hundred, and at 20 cm one part in two hundred and fifty.
EXAMPLE 21. ROWLAND'S FORMULA (20). FOR COILS OF EQUAL RADII BUT UNEQUAL SECTION
L,et us take #=25,^=10 as in the preceding example, but instead of supposing the sections of the coils to be equal let us take
o1 = 4 o % =2
Cx =I ^2=2
The values of d2 M and d M 2 ° %
8
°
will be the same as in the preced-
ing example. Substituting these in (20) we find JM= 0.6974^ M = 107.4885^ JM= 0.697477 M •'•^7 =Io8 l8 59 =339- - 7r 8 7 61 cm -
The correction here is fourteen times as great as in the previous example, although the areas of the cross sections of the two coils are the same as in the preceding case.
EXAMPLE 22. RAYLEIGH'S FORMULA (24). FOR COAXIAL COILS OF EQUAL RADII
^ = # = 25 £= 4 c=i d=io
We now find by formula (1) in accordance with formula (24) the mutual inductance of the following pairs of circles (Fig. 13):
O, 1 when ^ = 25, ^ = 25.5, d=io\ O, 4 when #=25, ^ = 24.5, d= 10; O, 2 when a A = 2$ and <f=8; O, 3 when A = a = 2$ d= 12 = )
and finally O, O' when A = a = 25, d= 10. Thus: 21674 — 12 4 46 Bulletin of the Bureau of Standards [ Fo/. <?, ^Vo. 7
' { > { ' c=l M = 109.3217^ 1
^=105.428777
^2=127.394977
< d=lo-
M=3 91.920677
434.065977
\a=25
J/ = 107.488577
326.57747T
,\M= 108.8 sgi'/r
.^ = 107.488577
Fig. 13
JM= 1.370677 cm.
EXAMPLE 23. RAYLEIGH'S FORMULA (23). COILS OF UNEQUAL RADII AND UNEQUAL SECTION Let ^ = 25 ^1=4 ^i =I <^= 10 <2 = 20 b = 2 2 ^2=3 We have then to calculate the mutual inductances of the following pairs of circles:
A a d A a d
M x 24.5 20 10 M 6 25 20 11
M 2 25 20 8 M, 25 21.5 10
M 3 25.5 20 10 M s 25 20 9
M± 25 20 12 ^0 25 20 10
Ms 25 18.5 10
se have been calculated by means of Havelock's fo rmu
le following results:
M = 248.41280
x
M = 2^.04027
2
M = A77440
3 ^= 248.7873
Af= 214.75755
M = 216.60185
b
M = 231.04386
6
M = 279.81417
7
M = 268.09410
s
Sum = 1996.5390
2 M = 497-5746
Diff = 1498.9644
2;
DifL = 249.8272 =
M
nx n%
—
Gwer] Formulas for Mutual and Self-Inductance 47
EXAMPLE 24. LYLE'S FORMULA (27). FOR COILS OF SQUARE SECTION A = a = z$ cm £ = <r=2cm d= 10 cm.
The equivalent radius r = a(i-\ 3 )
r=25 I+
( i^) =25 -°° 667 cm.
M is now found by using formula 1, 8, or 13, employing- r in
place of a as the radius. The result is M= 337.8475, agreeing very closely with the result found under example 20. M- M = 4M= .051 7tt
EXAMPLE 25. LYLE'S FORMULA (28). FOR COILS OF RECTANGULAR
SECTION
A = a = 2$ C= I d= 10
r=z4 I + — )= 25.OO167/
°
°V 15000/
b
2
— c = is
2
/3
2
= -^=1.25, 2/3=2.236 cm, the distance apart of the
two filaments which replace the coil (Fig. 14). We now find by formula (1), (8), or (13) the mutual inductances of two circles 1, 2 on the two circles 3, 4, where a= 25.00167 and d is 7.764, 10 and 12.236 cm, respectively. Thus: 2M = 215.0022877 13
^u= 90.31304^ M= 23 130.1406077 -2£-x- •7^7-6/r- — -5-e 2/J >j
\M =435.4559277 M =108.8640 .*. 7T
M = 107.4885 7T -1272S6-cm-
Fig. 14
JM = 1.3735 7T
^ J/" = the correction for section of the coils whose dimensions are given above. These values of M and JM agree nearly with the results obtained in example 22 above. 48 Bulletin of the Bureau of Standards [Voi.8,No.i
EXAMPLE 26. LYLE'S FORMULA (28). FOR UNEQUAL COILS OF RECTAN- GULAR SECTION Let us take the satne coils as in example 23 A = 25 K=\ ci =T d=io a = 20 ^2=2 ^2 =3 For trie first coil we find
r=25f 1 +
^2 j =25.001667 cm ^
•
/3 = 1.118034 cm
For the second coil
/ *
r=2ol 1 H ~ \1= 20.008333 cm
3= 0.645497
r+ 8= 20.653830
r— S = 19.362836
We then calculate the mutual inductance of the following pairs
of circles: A a d M ls 25.001667 20.653830 11.118034 Mu u 19.362836 11.118034 Mn a 20.653830 8.881966 Mu a 19.362836 8.881966
The results by Havelock's formula (16) were
^3 = 241. 29369
J/U =2l6. 91302
^3 = 286. 1349°
^=255.03471
Sum =999.37632
1
Sum = 249.8441 =
M
4 *i*s
which differs from the value by Rayleigh's formula (23) by six or seven in a hundred thousand. A more accurate value would, in each case, be found if each coil were subdivided and the formulas applied to each of the components as described on page 43. Such a proceeding is, however, rather tedious, although necessary in precise work. Grover] Formulas for Mutual and Self-Inductance 49
EXAMPLE 27. ROSA'S FORMULA (29). FOR COILS OF EQUAL RADII A = a = 2$ b= \ c==i d=io (same coils as examples 22, 25). &a 1 loge -j = log e 20= 2.9957 1
xtf
,
+c
9
96a*
2
-
-
log,— = ^-P
Be
Sa
d
—^^
49x2.9957 =
60000
.002446 °5 ^
,
£
2
-^ 15
.0125000
2
I2<^ I200
2^* + 2<f - 5^V 434 .0003617
120a?* 1200000
ifi* - $c« + i4^V - i4gV 8925
.0000177
504a?
6
^"504Xio 6
6^ + 6^ + 5^V _ 1622
.0000045
V 57600 ""360X10 2 6
2
jc*d / 80 163'
10240 <lo*?-f)
= -J*^ .0153322
lib 2 —**? 173 ^—
£- = = - .0014417
1920 120000 '
A
i§b2 d2 / 8a 97\
80
Se -7-^ )= -.0000827
J log. -.0015244
IO240*V
^ x 60/
7 <tf §-^
.0138078
\a = 100, .-. JM= 1.3808 7r cm.
This is a little larger value than found by formulas (24) and (28), and we shall see later that it is more nearly correct than either of trie other values.
EXAMPLE 28. ROSA'S FORMULAS (30) AND (31). FOR COILS OF EQUAL RADII AND SQUARE SECTION
A = a = 25 = b = C=2 d= 10
- 8a
log^-i = 2.9957-1 = 1-9957
2
iyb 68
72 .0028 1.9985
24O0 24000
-a b 2 2
2500 - .0500
50000
300x1.6624
- -o499 -.0999
1 0000
1.8986
50 Bulletin of the Bureau of Standards \Voi.8 No.i y
6a ° °
150
The approximate formula (31) would have given .0519 (agreeing with formulas 21 and 27), which would be amply accurate for any- experimental purpose. When the section is larger these small terms are, however, more important.
EXAMPLE 29. SECOND EXAMPLE BY FORMULA (30)
A = a=2$ b = c=$ d=io
8a
,
ge T = I ' 9957
~d~
2
17 b
-L r
2
= .OI77 2.0I34
240^'
-a b 2 2
.
">24
^( ^i-i)=-^9
lo
1. 6510
6a 150
.'.JM=0.27$2iT
M= 107.488577 (see example 20)
M= 107.763777 cm.
n. n Q
This is a very simple formula for computing JM, and within a considerable range (i. e., d not larger than a and yet the coils not in contact) it is very accurate.
EXAMPLE 30. FORMULA (32). COILS OF EQUAL RADII, BUT UNEQUAL SECTION For this we will take the coils of example 21 # = 25 d=io ^=4 £2 = 2 c,= x c9 = 2, Grfver] Formulas for Mutual and Self-Inductance 51
= 0.0016227
1st term
2d " = —0.0008542
3d " = 0.0062500
4th " = 0.0000570
Sum = 0.0070755
.\JM= 0.70755^
M = 107.4885^
Sum = 108.1960^
M
...—=339.9078 cm.
This example shows that the fourth differentials neglected in (20) here amount to one part in ten thousand.
EXAMPLE 31. ROSA-WEINSTEIN FORMULA (33). FOR COILS OF EQUAL RADII AND EQUAL SECTION
a = 2$ &=4 c=i d=io
. 2500 25
^ = 15.0000533 sm 2
^=ig^=iS
tf2 = 0.0020267 00*7-5555=55
ar3 = O.2170000 2 = .0000667
a -flr8 -a8 + (2aa -3aa ) COS2 ry + 8ad 008*7=14.7587120
1
a + — + 2<x3 + (2a2 + 3a3) COS2 7 + 8^3 COS* 7= 15.4628292
1
2
A = 0.0004730 Also F= 3.0351168
.#=0.0123901 £"=1.0489686
(f-e)(a + ^ = o.oo,o 7 i 9
^^=0.0129968
Sum = 0.0140687
47m sin 7= I007T.-/55 .*. JM= 1.379577-cm,
\ 16
This is not as simple to calculate as (29) and when d is less than a\z is less accurate than (29). But for d= a or greater it is more accu- rate than (29), and indeed the most accurate of all the formulas. 52 Bulletin of the Bureau of Standards [ Vol. 8, No. i
EXAMPLE 32. FORMULA (34). MUTUAL INDUCTANCE IN TERMS OF SELF- INDUCTANCE. FOR COILS RELATIVELY NEAR
For a = 25, #= 1, c=i, we have, n being the number of turns in one of the two equal coils,
L = 4.7ran (4.103816)
1
2
For b = 2, c= 1,
L^ = \iran % (4x3.698695)
For £ = 3, c=1 ,
L = \iran (9X3.411766)
2
Then the mutual inductance of 1 on 3 is by formula (34)
5=3
2
M = \irari -L + L,-2L 1 2
CO lH C^1
'
30.705894 + 4- 103816 - 29.589 560
= ^Trari'
2 }
= 477Y*;z X 2.610075
2
a=25 = 819.979 n cm. 2
If ?z = 100,
M= 8.19979 millihenrys,
as the mutual inductance of coil 1 on coil 3,
Fig. 15
Fig- 15-
EXAMPLE 33. FORMULA (35). MUTUAL INDUCTANCE BY GEOMETRICAL MEAN DISTANCE -4 = 25.1 ^ = 25.0 b = c =0.1 cm d=o.\ cm. The geometrical mean distance of two coils, corner to corner, as r in Fig. 10, is 0.997701, and log — = 0.002302 jK
.'. JM= 100X0.002302 (l.002)7T
= 0.2307^ cm.
3. MUTUAL INDUCTANCE OF COAXIAL SOLENOIDS
There are several formulas for the calculation of the mutual inductance of coaxial solenoids. Although few of these formulas ^ :
G°rwer] Formulas for Mutual and Self-Inductance 53
are exact, trie approximate formulas often permit inductances to be calculated with very great accuracy by using a sufficient number of terms of the series by which they are expressed.
CONCENTRIC, COAXIAL SOLENOIDS OF EQUAL LENGTH
MAXWELL'S FORMULA 29
The mutual inductance M of two coaxial solenoids of equal length (Fig. 16) is given by the following expression, due to Max- well, where A and a are the radii of the outer and inner solenoids, respectively, / is the common length, and n and n the number of x %
turns of wire per cm on the single layer winding of the outer and inner solenoids, respectively M= /pian n [/— 2 A a] x %
where
r=^l + A 2 2
_ a~ A-r+l a*_/ A^_ a?(i A^_sA \ 7
64A\2 + r
2
zA i6A\ 3
r ) 5
2
7
r J
2048 A\j r
7+4 r 9 3
r 11 )
7
63 ^/5,64d!_,«^
4*
QQ^'_M3^
+55
"2.i28M\9 9 r 9 r 11 r13 ' 3 r 15 J
231 a / J-
-^——J 7
10
128 A + I 28 A^3--4io-T5
11
,A A A
+ 520—^-221 —
13 15 11 19
512 <A V 11 11 r rr
r r r r^ )
/
429 a 12 /2i 512 A
13
,
~~ 6 4°7T5+32oo-
A 15
A - Q
17
, A 19
2.1024 ^\T ^ y 2>
Jr
)
3
p 6800-^
>
-^i
23
- 2261 pnrj [36]
r
6435^7"
6435
8192
a u /ii
^V
5
io2
1024
J5
^
^
15
, Tr ^
^^
^
17
T , 00
QQ ^^^ 103360^
W9
,,
r
21
, ^ 208012 A 37145 ^ \
-54264^ +
23
^—^-^ ^J 25 27
Electricity and Magnetism, Vol. II, \ 678.
54 Bulletin of the Bureau of Standards [ Vol. 8, No. I
Putting
M=M -JM n
M= 4-7r
2
a2 n 1 n 2 l is the mutual inductance of an infinite outer
solenoid and the finite inner solenoid, while 4M is the correction due to the ends. Equation (36) is Maxwell's expression, except that we have car- ried it out much further than Maxwell did. We would, however, emphasize that in the great majority of cases only three or four terms need be calculated in a, and in these only the first few terms in each parenthesis, to obtain a satisfactory accuracy.
< z
>
\
A a
f
Fig. 16
Since, however, this formula is the most valuable single expres-
sion known for the case of solenoids of equal length, it has seemed advisable to extend the series far enough to take care of the most unfavorable cases, which may arise in practice. At the same time the extra terms found have proved of use in checking our extension of Roiti's formula below. It should be noticed that the algebraic sums of the coefficients in each of the parentheses is equal to zero. For very long coils
J — small J the quantities in the parentheses are sensibly equal to
the absolute term inside For very short coils the parentheses are
a little larger, reachinlo- a maximum m the region A — - = °-9> but fall-
ing abruptly to zero at the limit A — = 1. The expression for a is GroveA Formulas for Mutual and Self-Inductance 55
therefore rapidly convergent for coils of all lengths, even when the inner radins is nearly as great as the outer radius. In such cases the number of terms to be calculated in the above formula may become considerable, but even then it is simpler to use this series than to make the calculation with an absolute formula, such as those of Cohen or Nagaoka. Equation (36) shows that the mutual inductance is proportional to /— 2 A a; or the length / must be reduced by A a on each end. When a is small and / is large, a is 1/2 approximately. That is, the length / is reduced by A, the radius of the outer solenoid. For the case of two coils each of more than one layer the above formula may be used, A and a being the mean radii, and n x and n 2 the total number of turns per cm in all the layers. The result will be only approximate, but usually less in error than if one uses the formula of Maxwell § 679 quoted by Mascart and Joubert. 30 When the solenoids are very long in comparison with the radii, formula (36) may be simplified by omitting the terms in A/l A^jr*, }
A jr etc. b h , The expression for a then becomes
1 a a* 50*
a= [37]
2 16A 2 128A* 2048A"
Heaviside 31 gives formula (2>7)i ^ ut as ^ neglects res an extension of formuh
A A etc., the additional terms are of no importance, being smaller V r ,
than the terms already neglected in (37)
HAVELOCK'S FORMULA 32
This formula for coaxial, concentric solenoids of equal length bears a close resemblance to the preceding, the main difference being that here / enters in place of the quantity r=^l 2 + A 2 in
30 Electricity and Magnetism, Vol. I, p. 533. 31 There are some misprints in Heaviside, 2, p. 277. The radius of the inner sole- noid should be c2y of the outer cu and p is c22/c12 .
32 Phil. Mag., 15, p. 339; 1908. There is a misprint in Havelock's equation (25). In the factor outside the brackets, read a 2 instead of a. 56 Bulletin of the Bureau of Standards \voi. 8, no. 1
equation (36). Using the same notation as in the latter this formula reads:
where
^_[~i 1 a* 1 a* 5 #6 35 a8
+
# 4+44Xt)'- + '•••] M
Havelock gives the expressions for the general terms in — and
— so that the computation of , /3 may be carried out so as to include terms of higher order when necessary. These expressions are
_ (2*-i)[i. 3 -5
m+1
(*"-3)Y («Y
2 n\(n + i)\ \AJ
and
(-iy( 2 s)\F^-s-i,-s, 2,^L
AY ;
S+1
2s+2
2 sl (s+i)\
a2
where F is. a hypergeometric series in —^ all of whose terms after
that in ( —- w 1 are zero.
\ >/->/> J -r
I>fy
T j. 3.7(7+1)
flr(flf+l)(^+2)j8(/g+l)(^+2)
l!2
^
.
3.7(7+1X7+2)
Formula (38) may be regarded as intermediate between (36) and (37), being applicable only to coils whose length is greater than the radius of the larger coil. In such cases, however, it furnishes a valuable check on Maxwell's formula. c°o°«-] Formulas for Mutual and Self-Inductance 57
CONCENTRIC COAXIAL SOLENOIDS, INNER COIL SHORTER THAN THE OUTER RdlTI'S FORMULA
For a pair of concentric, coaxial solenoids of which the inner ss solenoid is shorter than the outer, we have the following-:
Fig. 17
[39]
1024
693
2048 "
\
V 3^
3 ^4' ai ^4' Aft" ft
1
'/
+ 16384* A
K + AA> + 7 A< + 14 A'Kp? ft
15
/
in which (see Fig. 17)
33 For the derivation and method of extension of this formula see this Bulletin, 3, pp. 309-310. Recently we have carried it out still further to include the case of coils of moderate length. This formula was originally given (without proof and includ-
ing the main term in ( J-__L\ only) in this Bulletin, 2, p. 130; 1906. \Pi P2 / 58 Bulletin of the Bureau of Standards \voi. 8, No. 1
x—l
Pi = -y/^+A where l =
2
x
2
X+l
P,= ^TA' " 4=
/= /2 — lx = length of inner solenoid.
x = length of onter solenoid and A and a the radii.
When - is small (case of short inner coil), (p — p ) is most accu- 2 t
xl
rately calculated by the exact formula (p% — p^) = , the denomi- Pi+Ps nator being calculated from the above expressions for p1 and p2 .
For long coils ( — small
J
the above formula is rapidly conver-
gent, especially if the inner coil is considerably shorter than the
outer. This formula may also be used for short coils ( —-* small V
the convergence being most rapid when the radius of the inner coil is small in comparison with that of For very short coils, the outer. we have expanded formula (39) in a series in ascending powers of a2 -jr2 . This formula is, however, not so accurate, nor so simple to use as that of Searle and Airey, and has not been included in this collection. A peculiarity of Roiti's formula is that the successive terms, especially in the case of short coils, are nearly equal in pairs. Thus — A the terms in \Pi ( and [ —3 =
?V
7 )
P2J W
are of the same order of
% i
magnitude, but of opposite sign similarly for the terms involving ;
the ninth and eleventh powers of p± and p 2 and so on. For the ,
limiting case x = /, Roiti's formula goes over into Maxwell's (36), as
would be expected, since both are derived by integration of the same
original expression between appropriate limits. To obtain, how-
ever, the same precision, twice as many terms have to be calculated
in Roiti's formula as in Maxwell's. We see from these considera-
tions, that in using Roiti's formula, the inner coil need not be very
different in length from the outer coil, although in general the con-
vergence is better with a relatively short inner solenoid.
Rosa I
Grover
Formulas for Mutual and Self-Inductance 59
GRAY'S FORMULA
Gray 34 gives a general expression for the mutual kinetic energy of two solenoidal coils which may or may not be concentric, and their axes may be at any angle <£. The most important case in practice is when the two coils are coaxial. In that case the zonal harmonic factors in each term reduce to unity, and half the terms become zero. Putting the current in each equal to unity, the mutual kinetic energy becomes the mutual inductance M.
2x
*^Tzr_
n
-r^-^,o -*— AXIS
Fig. 18
Let 2x, A, n x be respectively the length, radius, and number of turns per cm of one of the coils, and 2/, a n % be the corresponding }
quantities for the other solenoid. Let, further, x and x x % be the distances, along the axis, between the center of the coil with radius a and the nearer and further end planes, respectively, of the coil with radius A^smd let r± and r2 be the diagonals (Fig. 18).
r1 =^/x1 2 + A 2
r2 = ^Xz * + A*
Gray's expression with these changes becomes
M= ir'a'A'n.n, \K k + K k + K k +x x z z h h ] [4o]
where K A" x, 3, etc., are functions of x and A, and k k x, z, etc., are 35 functions of / and a.
34
Absolute Measurements, 2, Part I, p. 274, equation 53.
35
Rosa, this Bulletin, 3, p. 221; 1907.
,
60 Bureau of Standards Bulletin of the Bureaii \voi. <?, Noi
2 (x t x\
i/x l
x,\
x
^-M-^- ^-4)} 1
•X'l
y I
x y
% / ! 1
[40a]
/4 f r ^r
1 '2
£j = 2/
^s=-^3-4|)=-«VA
\2 a a /
\i6 2 # «* ^ay
This formula is simple and convenient for calculation, if only a few terms need be evaluated. This is the case when rx and r2 are large (coils relatively far apart). The coefficients L2n) X' in and X" m I 2 x 2
are derived from the same polynomial S2n by substituting —2 -^ a A x 2
and -~, respectively. A For short coils relatively far apart these polynomials are all small. Table XVIII gives values of the polynomial Sin with varying argument, to aid in calculations where great accuracy is not desired, or to aid in making preliminary calculations to see whether the convergence will be satisfactory in any particular case. If the coils are concentric, and the ratio of the length of the
winding of the outer coil to the radius is -^3 to 1 b = o, and if the , K same condition holds for the inner coil, k 3 = o. If in addition a is considerably smaller than A, the terms of higher order become
negligible and (40) reduces to ..-. 27rViV*-? iV2 M=- r — 1 [41] -. GrTver] Formulas for Mutual and Self-Inductance 61
where d is half the diagonal of the outer coil, = -yJx^ + A When 2 .
the dimensions depart slightly from these theoretical ratios the 3508 small correction terms to (41) can be calculated. The general case for concentric coils is treated in the next section.
SEARLE AND AIREY'S FORMULA
The following expression for the mutual inductance of two con- centric, coaxial solenoidal coils (Fig. 19) has been given by Searle and Airey: 36
M=g G +g- G +g G* --giGi + W; A x x 3 3 6
27T #
2 2
4/
2
- 3a 2
A\4x - $A ) SI' - 20/
2 2
V+ 5a*
2d' 8d> 8
A\Sx' - 2ox2A + $A") (64/ 6 - 336/
v2
2
"
V + 280/V - 35a 6
)
i6d 64 ]
[42]
The notation of (42) differs slightly from that used by Searle and Airey.
Fig. 19
Equation (42) has been extended and put for greater convenience in calculation into the form 37 shown on next page.
35a Rosa, this Bulletin,
3, p. 221; 1907.
36
The Electrician (I^ondon), 56, p. 318; 1905.
37
Rosa, this Bulletin, 3, p. 224; 1907.
21674 — 12 5
62 Bulletin of the Bureau of Standards iVoi. <?, No. i
M - 27rWN JVf 1 Aa T2 2 A' a* T
t"
AW712^CA + A* a 6^6 A + 1
- - u
3 2^
12 6
3 2^ *
+ ^(^?)^- A.+ 2 • • 1 [43]
where
A=3~4T? d-J^ + A*
-^2 "
= 3 ~ 4^8
v 5 # 2
,
X* A = -- >-2 + 4-i
~*~2 ^#
I( 2 4
tf
Z/'6 _35 - 35
l% ? f
3 4
TP + 2I ? _4 ?
x _35 35* ,
* *" ~i6
T 63 105/ 2 /
4
,/ 6 /»
32 4 ^ 2
*^
,. _23I II55/ 2 1155^* r I*
-36^ + 4^ - 128 32 *' 8 «
J^ = 2^^j and iV — zln % are trie total number of turns on the two 2
solenoids. This formula reduces to (41) when the terms after the first are negligible, as they are when the conditions assumed for (41) are fulfilled. The above expressions for Z X show what these 2, 2
conditions are in order to make the second and third terms zero. 2 2 If I 1 a is slightly more or less than ^, (43) gives the value of the second term which is neglected in (41), etc. The degree of convergence of Searle and Airey's formula depends primarily on the magnitude of the quantity A a — *- 2 2
in certain cases,
;
however, the values of the coefficients become of equal importance, making it necessary to examine carefully into the degree of con- vergence of the formula, since the terms of higher order are some- times larger than those immediately preceding. Since the X and L GrTve?] Formulas for Mutual and Self-Inductance 63
P x z
coefficients are polynomials in —^ and -^, each one will have a finite a A number of roots depending on the degree of the polynomial. The / x values of these coefficients will therefore, with increasing - or -j a /i ,
between positive and negative values, each maximum or
oscillate minimum being greater than that preceding, until, for values of the argument greater than the largest root, the values of the functions increase indefinitely without limit.
For short coils ( -^ and - small J
the coefficients will evidently
be confined to moderate values, and if, further, the inner radius is small relatively to the outer, the convergence will be very rapid. For longer coils the coefficients may attain very large values, and the convergence become very unsatisfactory, in spite of the fact that A z a? —-£- is, for given radii, smaller with long coils than with short coils.
The conditions are so complicated that we have calculated (Table XIX) certain values of the coefficients to aid in deciding whether, in any given case, the convergence will be satisfactory or not. The x I values given for -^ and - less than unity will also be found useful aJl in calculations of the mutual inductance of short coils by Searle and Airey's formula, when the highest precision is not required. Coeffi- cients of higher order than those given above are calculated by the formula
P=* (- i) n -p (2n + i)2n(2n - 1) \2n-{2p — 2)~\/lV n ~ 2p
W
• • • •
L n== //+ I \ 2 2. * 2j 4 2. 6 2 (^ X I Xm is calculated by the same expression in -^ instead of -• A a Table XVIII includes all the positive and negative maxima as well as the zero points of the coefficients up to and including Z u or A"u together with the values at a number of intermediate points. ,
Although, from the nature of the case, a table to serve as the basis of accurate calculations would be somewhat bulky, those given should suffice to simplify the use of this valuable formula. a
64 Bulletin of the Bureau of Standards [Voi.8,No.i
COHEN'S FORMULA 38 FOR ANY TWO COAXIAL, CONCENTRIC SOLENOIDS
This is an absolute formula for two coaxial, concentric solenoids of lengths 2lx and 2/2 Fig. 20. ,
M^^jT7i nly-V^)
x
V=-(A*- 2
)c[E{F[k\ 0) - E(k', 6)} - EF{k f
, $)]
+
c'~ (A* - 6Aa + a 2 y - 2 (A - aj F 2
'
[44]
3-yl(A + a)* + i*
2{A * + * 2) ~
+ ^ + ay +
(A ?. 2T- <^ - «)
F is obtained from Fby replacing ^ by ^, x
i? and ^ are the complete elliptic integrals of the first and second kind to modulus k, where £ 2 = 4 ^ F(k', 0) and £*(£', 0) are the incomplete elliptic integrals of modulus k! and amplitude 0, 2L
27, £' 2 =I-£ =I_
2 \Aa
r
1
I 1
(A+af + 4*
1 J— -4—
(A + a)* + c*
1
A {A 2 -a 2 y + cf(A+a)
Fig. 20
NAGAOKA'S FORMULA FOR ANY COAXIAL SOLENOIDS
Nagaoka has recently given 39 an absolute formula for the mutual
inductance of two coaxial solenoids, whether concentric or not, and
38 This Bulletin, 3, p. 301; 1907. 39 There are a number of misprints in Jour. Coll. Sci., Tokyo, 27, art. 6; 1909. Nagaoka's article, which we have detected and corrected by a careful check on the derivation of the formulas. Grller] Formulas for Mutual and Self-Inductance 65
has expanded this in q functions in a form suitable for calculation. In the notation of Fig. 18,
M= \irn n^Aa(I -/,-/, +
x x 1,) [45]
where Ily /,, 73 and Ik are the values of the integral /, given below ,
with the arguments c=d+ (x -f /), d+ (x—l), d— (x — I) and d— (x + /) respectively, where d is the distance between centers. The expression for / is, in the Weierstrassian notation, •
/- 3 - P^x + P^+^^-aV^)))
[(§
where v is an auxiliary quantity, and g^ and g% are respectively the real semiperiod and invariant of the Weierstrassian function j)u.
To calculate /, Nagaoka divides it in two parts
We then calculate the following auxiliary quantities
^ KVv e^- 2Aaa
(A + af
P> = (Vv-e*) =
2Aaa
and thence (e — e2 ) (ex — 3 ), and (ez — e3 ), which with the relation x y <?
e e = o enable us to find pz/. ( i + 2 + 3) e The very small quantity q is found, as in formula (8), by the relations (see also Table XV)
9
<-H0'"<0 +
b} ==
^2~^3 b>* _ ^1~^2 7_ 1 ~~ V k
*i-*3 *i~e9 i + Jk' (i + £')(i + V*')'
66 Bulletin of the Bureau of Standards _Voi.8,No.z
and Wj may be calculated by any one of the following equations (the other two serving as checks):
4-\ex ez
The term F is now given by either of the following two formulas,
which give a check on the calculation:
/' - _ B(5l+A) j. {P. + zP^-e^ _ p^ ^>)
+
1 2 6 p 4^(0) "
/' = PJA±P$ _ (P + 2P )(e -e ]„ _ p^ g,"(o)
\ 1 s 2 s)
I 2 6 P 4«,«.(o)
The quotients of the 6 functions are easily calculated from the known value of q and the relations
fl,"(o) __ My + 4g' + 9g'+ • • •)
3 (O)
I + 2? + 2/ + 2? + ....
9
o "(o) _ 8^-4/ + ^- • • • •)
O (O)
I -20+ 2/- 2^+ • • • •
To calculate /" we have
7" -
-^^^^[V^ + ^^{* -««- *)}]
the expression in the brackets being nearly equal to unity. The quantity b is calculated from the equations
— £ = cos 2irw = — (i + 2<7* COS /\.7TW+ ' - •)
7
Grover
Formulas for Mutual and Self-Inductance 67
first putting cos 2irw equal to its approximate value — and then ,
computing the small correction in cos \ttw from cos 27rze>, remem- bering that w is a pure imaginary. The correction to b thus found is often negligible.
The term I" becomes less important as the difference of the radii
of the solenoids becomes small, and vanishes for equal radii. If, further, the lengths of the solenoids be equal also, I2 = 73 and we ,
have only three of the integrals to evaluate, and only the first term /' in each of these.
For concentric, coaxial solenoids <^=o, and consequently
/ — x 2 = / -/ 4 3 , so that only two integrals must be calculated. On account of the number of auxiliary quantities involved, Nagaoka's formula should not be employed except when the various series formulas given in this section are all shown to be inadequate. It is, however, simpler to use Nagaoka's formula than the elliptic integral formula from which it is derived, or any other expression in incomplete integrals yet derived, even supposing L,egendre's table of incomplete integrals to be available.
RUSSELL'S FORMULAS 40
Russell's formula for coaxial solenoids in the notation of this
paper is
M= 47rV« 4~^ 1 1 |i -\q % K - ^ k ~ J^j%& i
2.4.6.8^ x 2.4.6.8. io^
1
^?Ji q % k£ --q^i ~ terms with above coefs.l [46]
where ^Aa i?1 3 = (^+^) + (/ + / ) 2 1 2 2 k^ =
^Aa
~Rj
Alexander Russell, Phil. Mag., Apr. 1907, p. 420.
/
68 Bulletin of the Bureau of Standards [Voi.8, No. i
(A+a) 2
i 1.3.5. .-.2«-3 ^
^n = \Aa ^ n x
2n — 2 a
7224.6
(A + af 11 A
H%
\Aa 22a
(A + af 11.3A
Hz
4Aa ^2 32.4a
etc.
A and a are the radii of the outer and inner cylinders respectively, 2 lx and 2 2 their lengths, Fig. 20, and #„ n % the number of turns of wire per cm in the two windings. This formula applies only when the inner coil is shorter than the outer.For two coils of equal length the second part of the above formula is not convergent, and hence it must be replaced by an expression in elliptic integrals. The formula thus becomes (equation 42 in Russell's paper)
M= 47r a n nl R
2 2
1 x
1 - -q k - ^q9 k* —
%
2
• • • as above!
+ ^^^i{A + a )(F-E)-2AaF-] 2 2
[47]
the modulus of the elliptic integrals being k % = —^
This formula gives an accurate result for equal solenoids of con-
siderable length, but Maxwell's formula (36) is just as accurate and much more convenient. For short coils neither (46) nor (47) will apply, but for that case as well as other cases Russell's general formula may be used. As the latter is equivalent to (44) it is not here given.
MUTUAL INDUCTANCE OF A SHORT SECONDARY ON THE OUTSIDE OF A LONG PRIMARY This is an important case in practice. Havelock n has shown that the mutual inductance of two such solenoids is the same as that of two coils with the same radii and lengths, but with the shorter coil inside. That is, the mutual inductance of a coil of length / and radius A outside of a coil of length x and radius a is the same as
41
Phil. Mag., 15, p. 343; 1908.
:
GrTver ] Formulas for Mutual and Self-Inductance 69
the mutual inductance of a coil of length x and radius A outside of a coil of length / and radius a. The series formulas already given for the latter case may there- fore be applied to the present case directly if the quantities / and x, or lx and /2 be interchanged. ,
In Roiti's formula we put, therefore, lx = instead of 2 2 The values of p and p are, however, unchanged and the formula x 2
may be used just as it stands. Russell's formula being symmetrical in lx and /2 requires no change whatever. In Searle and Airey's formula we have to put
d^^JP + A'
x 2
A = 3-4-,
%
l
Z = 5-io^ + 4^
4
2 a2 ^a*
etc.
Cohen's and Nagaoka's formulas apply without change as would be expected.
ROSA'S FORMULAS FOR SINGLE LAYER COILS OF EQUAL RADII
The mutual inductance of two coaxial single layer coils of equal radii is given by the following expression
M
N N = M + 4M
,, ,,,,
X %
where M is the mutual inductance of the two parallel circles at the
centers of the coils and JM is given by the following expression: iZ
42
Rosa, this Bulletin, 2, p. 351; 1906.
«
7o Bulletin of the Bureau of Standards [Vol. 8, No. I
4M= \ira 24 3* <^i-f)-f£{^-£) t
d' 8 a"
1050^/ 8a_ 54 \ 44100^/ 8a 37 93 \
=
+
7rt a\
2 g d
35/ W^\ Z ~d 2^20;
+
10
4- —4
320a?
ti
"* + 16 a*
'
256 a\ °^ d 60/ [48]
2ioogA gg_gg3\
2 6 8
i28 « \ rf 420/
"*"
. K + V + 7(W: + *M) L , 3 <?_ 3 <? .
2688^ } i6oa 8 1024
+
(V + ^ + iaCW + VO+^Wj I+ x ^
18432a78 |
H2? +
For coils of equal breadth and equal radii (Fig. 21) b = b % = b and A
we may write the equation (48) as follows: PQOQ^OOQft OOOQ(j)OOOQ
Fig. 21
+
35(^l-g)-«»?(^?-iS) +
I ^ 15 a?/
log
8a_i87\
[49]
6oa? 16 a 8
256^V ^ 60/
,
aioo^A 8«_893\_
8 6
i^8 tf V * <* 42o/
1 b
+ z*s-ali
168 a?
+ 160 a
6 2
^ '
1024
— -
G°roveA Formulas for Mutual and Self-Inductance 71
This expression will give a very accurate value of JM for two coils not nearer together than their breadth if a is considerably greater than b, the breadth of the coil. For coils which are not so near together the Rosa-Weinstein for- mula * 3 may be used. JM=47ra sin 7 [F-E) P+EQ\ [50] where
P= COS ty
j-jl Oi - « - 3^3 cos 7 + Sa cos 7]
2
3
4
@ = 1Z5* ^+ 2afs
*
+ 3 ^ cos 7 + 8 ^ cos 7^
3 3
*
sm 7 = —J
z
1
—
\a % + d 2
-Tg cos' 7 =
'
—-s
4a 2 + d~
-j9
and i^and E are the complete elliptic integrals of the first and sec- ond kinds with modulus k = sin 7.
When the coils have equal breadth b = b = d and a = 3 a = — 1 2 x 2 , z 8 .
If the lengths of the coils are not very small in comparison with d a greater precision may be attained by adding to (50) the last two terms of (48) or (49) which depend on differentials of the sixth and eighth order.
MUTUAL INDUCTANCE BY MEANS OF SELF-INDUCTANCE FORMULA The mutual inductance of two coils having the same radii and the same number of turns per unit of length may be calculated with great accuracy from a knowledge of several self-inductances. If the two coils be designated as A and B and a coil C having the
same radius and number of turns per unit length be imagined to exactly nil up the space between A and B, the self-inductance of coils A, B and C in series will be
^abc = L* + Z B + LQ + 2MAC + 2 MBC + 2MAB
43
This Bulletin, 2, p. 342; 1906.
72 Bit lie tin of the Bu rea u of St a n da rds [ vol. 8,no.i
Similarly the self-inductances of the coils A and C in series, and of B and C in series are given by the equations. LAG = LA + LC + 2MAG LBC = LB + L C + 2MBC Eliminating M and M AC BC in the equation above we find M = (Z 2 AB ABC + Lc ) - (LAC + Zbc) [51] The self-inductances may be calculated with all the accuracy desired by Lorenz's or Nagaoka's formulas. Formula (51) is of especial value in testing new formulas and in the case where the two coils are in contact. In the latter case the formula becomes
2MAB = LAB - [LA + LB ) [52]
OTHER FORMULAS
Himstedt has given several formulas for different cases of coaxial solenoids. The first** is for the case of a short secondary on the outside of a long primary. The formula is very complicated, and the calculation tedious. The formulas of Roiti and Searle and Airey may be used to much better advantage. Himstedt's second expression is for the case of two coaxial sole- noids not concentric, the distance between their mean planes having any value; the radius of one is supposed to be considerably smaller than the other. This also is a very complicated formula, involving second and fourth derivatives of expressions containing the elliptic integrals F and E. Gray's general equation is much simpler to calculate. This is not, however, an important case in practice, and we do not therefore give Himstedt's equation. Himstedt's third equation is general and applies to two coaxial solenoids of nearly equal or very different radii, which may or may not be concentric. This expression of Himstedt's consists of four terms, each of which is a somewhat complicated expression involving both complete and
incomplete elliptic integrals. A less inconvenient general expres- sion for Min elliptic integrals is given above (44). Havelock* 5 gave a formula for the mutual inductance of two coaxial, concentric solenoids, which resembles somewhat the formula
44
Wied. Annalen, 26, p. 551 1885.
;
45
Phil. Mag., 15, p. 342 1908.
;
p
Rosa Grover. Formulas for Mutual and Self-Inductance 73
of Roiti. It is, however, not so convergent as the latter, except when one coil is very short in comparison with the other. After the present work had gone to press a valuable article appeared 46 * by Olshausen, in which the author derived a general absolute expression for the mutual inductance of two coaxial solen- oids. Adopting the same nomenclature as in Nagaoka's formula (45) , the integral / is in this case given by
I=m^2Aa\ (^- — 2 ^ W + pz 1 , -'?i + — Ui^ — ©1— (z^) + nm\ [52^]
Here m is a parameter, which is to be arbitrarily assigned, and consequently (520) may be put into various special forms depend- ing on the value assumed for m. The integer which may be ?z,
positive or negative, enters because of the many-valuedness of a logarithm, and is to be found from the equation denning v.
If we place m = I -^- Y and let n = o, the result is Nagaoka's equa- tion (45). The author shows further that by expressing the quanti- ties in (52a) in terms of the elliptic integrals of L,egendre, Cohen's absolute formula (44) may be shown to be a special case of the gen- eral equation (522). As a third example, the author shows that the absolute formula 5b of Kirchhoff, published for the first time by Cofnn in a form sub- 450 sequently shown by Cohen to be in error, is included in (52a), and the correct expression is given in Olshausen's equation (38). (A + a) 2 + c2 Olshausen showed further that if the value — J- be assigned & 2Aa to m, the expressions for some of the auxiliary quantities become very simple. For the details of calculation as arranged by him we refer to the original article.
CHOICE OF FORMULAS
i. — Coaxial solenoids, not concentric. (a) For the general case, if the greatest precision is required, Nagaoka's absolute formula (45) should be used. Since, however, the mutual inductance of such
45a Phys. Rev., 31, p. 617; 1910.
45b This Bulletin,
2, p. 125; 1906.
45c
This Bulletin, 3, p. 301; 1907.
74 Bulletin of the Bureau of Standards [Voi.8,no.j
coils will not in general be needed with extreme accuracy, it will usually be found sufficient to apply Gray's formula (40), taking the precaution to determine by a rough preliminary calculation, whether or not the terms of higher order will have an appreciable effect in the case at hand. For this purpose Table XVIII will be found of material aid. If the convergence is not satisfactory, or if more than three or four terms must be calculated, it will be found advantageous to sub- divide one or both of the coils, and to apply Gray's formula to the calculation of the mutual inductance of the several pairs of sections; for these the convergence will be more rapid. For example, if coil A be divided into two parts, C and D, and the coil B into sections B and F, then the mutual inductance of A or B will be given by the relation M = M + M + M^ + M AB CB CF DF
It may be stated as a general criterion for the rapid convergence
of Gray's formula, that the distance between the coils should be great relatively to the radii, and that the coils should not be very long. With long coils it is necessary to carry the subdivision further than with short coils, with a corresponding increase in J:he number of terms to be calculated, but even then the labor will gen- erally be much less than in using Nagaoka's formula. If the coils be relatively far apart, and great precision is not desired, the formula of quadratures (23) may be adapted to this case, by making the radial dimension of the cross section of the coils in Fig. 4 equal to zero. We have then
and the formula of quadratures becomes
It is, therefore, only necessary to calculate, by an appropriate formula or formulas, the mutual inductances of the five pairs of circles, and to take the weighted mean indicated. This formula is more accurate, the shorter the axial lengths of the solenoids in comparison with their distance apart, and the process of subdivision above described will be, in general, necessary. Gray's formula is, however, to be preferred. G°roveA Formulas for Mutual and Self-Inductance 75
(b) An important case in practice is that of solenoids of equal radii. If the coils be in contact or very near together the formulas
(52) or (51), respectively, should be employed. If the solenoids be separated so that the distance between their medial planes is greater than the axial length of either, the mutual inductance may be calculated from the mutual inductance of the two circles at their centers, a correction being applied to take account of the lengths of the coils. For this purpose formula (48) should be used for coils relatively near together and (50) for coils farther apart. The corresponding formulas, for coils of equal radii and equal length are (49) and (50). 2. Coaxial, concentric solenoids of equal length. — the solenoids If
be long relatively to their radii, Havelock's formula (38) will be found to be very accurate. Maxwell's formula (36), however, is applicable to both long and short solenoids, provided the radii are not too nearly equal, and should be given the preference, using Havelock's, when desired, as a check on the result. It may be
necessary in rare cases to use the absolute formulas of Nagaoka or Gohen. One should also bear in mind that Roiti's and Searle and Airey's formulas also hold for equal length solenoids, and may be used in checking the results. 3. Coaxial, concentric solenoids —Inner coil the shorter. —For relatively long coils Roiti's formula (39) will give very accurate values, whatever the length of the inner solenoid, provided the radius of the inner coil is not closely equal to that of the outer. Roiti's formula is also applicable to short solenoids in case the inner radius is considerably smaller than the outer. For short solenoids, however, Searle and Airey's formula (43) is preferable, and gives a very rapidly converging value unless the inner radius be nearly equal to the outer. Russell's formula (46) is most convergent for long solenoids, of which the inner one is a good deal shorter than the outer one. In those cases for which none of the above formulas converge rapidly, and great precision is desired, Nagaoka's or Cohen's abso- lute formula should be used. 4. Coaxial, concentric solenoids — Outer coil the shorter. —The formulas of the preceding section are to be used interchanging x and /, or lx and /2 as the case may be. The formulas of Roiti, 76 Bulletin of the Bitreau of Standards [Voi.8 No.i t
Russell, Cohen, and Nagaoka are unchanged in form; Searle and Airey's is slightly changed as regards the coefficients L and X. Usually, it will be found that more than one formula will apply to a given case. The advantage of such a check Can not be over- estimated. For illustrations and tests of the above formulas, see examples 34-47- In taking the dimensions of coils where an accurate value of the mutual inductance is sought it should be borne in mind that the above formulas have been derived on the supposition that the cur- rent is uniformly distributed over the length of the coaxial solenoids; in other words, these formulas are all current-sheet formulas. Hence, for coils made up of many turns of wire we must meet the conditions imposed by current-sheet formulas. In calculating self-inductances, this requires an accurate determination of the size of the wire and of the distance between the axes of successive wires, from which we can calculate two correction terms to be combined with the value of L given by the current-sheet formulas. 46
In the case of mutual inductances, however, there are no correc-
tion terms to calculate; but we must take the dimensions of the current sheets that are equivalent to the coils of wire; that is, the radius of each coil will be the mean distance to the center of the wire, and the length of each will be the over-all length, including the insulation, when the coil is wound of insulated wire in contact, or the length from center to center of a winding of n + 1 turns, 47 where n is the whole number of turns used. It is also very important that the winding on both coils shall be uniform, 48 and that the leads shall be brought out so that there shall be no mutual inductance due to them. The mutual inductance will of course be different at high fre- quencies from its value at low frequencies. We assume, however, that for all purposes for which an extremely acturate mutual induc- tance is desired the frequency of the current would be low, say,
46
Rosa, this Bulletin, 2, p. 181; 1906.
47
Rosa, this Bulletin, 2, p. 161, 1906; and 3, p. 1; 1907.
48
Searle and Airey, Electrician (I/ondon), 56, p. 318; 1905.
—
G°rovei\
Formulas for Mutual and Self-Inductance yj
j
not more than a few hundred per second. If the value at very high
frequency is desired the coil should be wound with stranded wire,
each strand of which is separately insulated.
EXAMPLES ILLUSTRATING THE FORMULAS FOR THE MUTUAL INDUC-
TANCE OF COAXIAL SOLENOIDS
EXAMPLE 34. MAXWELL'S FORMULA (36) AND COHEN'S (44)
Two solenoids, Fig. 22, of equal length, 200 cm, each wound with
a single layer coil.
-< £=200 »
A
10 f„
Axis
.
V . . .
Fig. 22
A = 10 = radius of outer.
a = 5 = radius of inner.
Substituting in (36) for a we have the following:
a 0.487508 - £ * (0.999875) - JL * (o. 5 ooooi) - ^|_
^ (I)
= O.487508 — O.OI5623 — O.OOO488 — O.OOOO38
= <M7 I 359
.*. V
M= Aifa (200 — 9.42718)
M= 19057.287J-V
IOO 7T 2 X IQO57.28,
xr
If n=io turns per cm, M=
1^
f-^ henry
= 0.01880878 henry.
The effect of the next term in the series for a beyond those cal-
culated is to raise the value of M by only one part in five million.
By the approximate formula of Maxwell (^y)
1 1 1
2a = 1 —
8x4 64X16 1024x64
= 0.96773
*. ^=0.018784 henry.
21674'
:
78 Bulletin of the Bureau of Standards Woi. 8, No. /
This example by Heaviside's extension of Maxwell's formnla (see
p. 55) has exactly the same valne of M; that is, the added terms do not amount to as much as a millionth of a henry in this par- ticular case. To show that the result by Maxwell's formula (36) is very accurate for this case we may now calculate M by Cohen's absolute formula: M=^irn\V-l) Substituting in (44) for Vwe have the following terms:
V= 7863.79 + 4200532.04 - 4169106.25 - 23561.95
= 15727.63
Vx 1392.18 — 632.16 = 760.02
=
.-. M= 47m (15727.63 — 760.02) = 47m 2 (14967.61)
2
= i9°57-3 67rV
J/= 0.01880886 henry.
This agrees with the result by Maxwell's formula to within five parts in a million, the value by Maxwell's formula being more nearly correct, as is shown in the next example. The example by Cohen's formula illustrates the disadvantage of that formula for numerical calculations. Aside from the fact that it is complicated, and involves the use of both complete and incom-
plete elliptic integrals, the value of M depends on the difference between very large positive and negative terms, so that in order to get a value of M correct to one part in one hundred thousand it is necessary in the above example to calculate the large terms to one part in twenty-five million. As a means of testing other formulas, however, this absolute formula is of great value.
EXAMPLE 35. HAVELOCK'S FORMULA (38)
We will take the same problem as in the preceding example #=5 A = \o 1=200 1 1 a2 2~ T6A°= a48 « 75 128 - —a=— 0.000488 A 1 ^ 1 Grover\ Formulas for Mutual and Self-Inductance 79
*6 _
5 — 0.000038
2048 A 6
1 A — 0.012500
4 /
1 / a*\A 3
+ 0.000010
16 V *
A) I 2 s
Sum = fi = 0471359
which is exactly the same as the value of a found by Maxwell's formula in the preceding example. The value of the mutual induc- tance agrees, therefore, exactly to seven significant figures with the value given by Maxwell's formula. For this example, accordingly, we see that Maxwell's and Havelock's formulas give a more accu- rate value than Cohen's formula, unless the quantities in the latter are carried out to a greater number of places of decimals. This was 49 pointed out by Havelock.
EXAMPLE 36. MAXWELL'S FORMULA (36). FOR EQUAL SHORT
SOLENOIDS
«=5 -^ = 10 1=2
r=VIo4 ^ = 0.9805808 OLJL
A-r + l
0.09009805
2A
1 a2 /
L
A\ 3
- 0.00089271
i6A\ r) 3
1 aVi A^ 5 A'\ - 0.00013073
64 ^\2 r5 2 r1 )
- -^(0.080378)-^-=
0/ '
v
-- 0.00002146
2048 A"
63 ,.
N a8 --
~ (a5 ° 79) 0.00000381
2 .lf8* A' ~
- =§ (o-788) £-- -- O.00000068
" ^
429
2.I024
,
(M3)
.
AAX*
a 12
~ ~- 0.00000012
Sum = a = O.08904854
49
Phil. Mag., 15, p. 341.
8o Bulletin of the Bureau of Standards [Vol.8,No.i
2Aa = I.7809708
l-2Aa= O.2190292
.'. M 17.20251
^Trn x n %
The formula is not so favorable in this case as for long coils, since the quantity 2 A a is nearly equal to /. Further, the quantities involved in the parentheses are rather large, although their sum is in only one case greater than unity. There is, however, no diffi- culty in obtaining these factors with all the accuracy required. We have carried out the calculation with this formula further than would in practice be desired, in order to test the formula. We find that to get the same order of accuracy by Searle and Airey's formula X terms including the product 12 LU must be calculated. The result found was M = 17.20252 or only one part in two million different. We have also calculated the mutual inductance of these coils by Nagaoka's formula and obtained a value not very different, but this is a very unfavorable case for this formula, no great accuracy being obtainable using seven-place logarithms.
EXAMPLE 37. ROITI'S FORMULA (39) COMPARED WITH SEARLE AND AIREY'S (43)
We will now calculate the example, Fig. 23 (originally given by
50
Searle and Airey ), by Roiti's formula, and also by the formula of Searle and Airey.
-2X-lQr
/
"^-4^4^ 1
4=5
p^\ cr=4i
1
22 —5 1
Fig. 23
30
Electrician (London), 56, p. 319; 1905.
G°roveA Formulas for Mutual and Self-Inductance 81
30 cm = length of outer solenoid.
5 " = " " inner "
A= 5 " = radius " outer "
a= 4. " = " " inner "
N = 300 turns
x
.*. nx =- — = 10 per cm
N = 200 "
,
r
%
?z
2
= 200 = 40 per cm
o
A = I 2.5 ft = Vi275 + 25 =
2
1 3462912
4 =I 7-5 ft = Vi7^5 + 25 =
2
18.200275
A ft --ft = 4-7373 6 3
P1 + P2 = s1 6 18 ?
-
^
It is more accurate to calculate (p — p^) by the formula %
xl
This gives (/o 2 — /°i) = 4-7373^20, which value will be used in the calculation of M.
ft-ft= 4-7373 620
AW
- w-s )- + 0121975
1
-
^V/ 1 1 \
^v i+
64 ( ^X?-s) =+ oooi8°8 -
35^ V/
5 ia Vi
2
i\
256
xo$A*d / Q a2 1 a* \ / 1 1 \
1024
6g^A aV 6
/ I+ 2_^
2 <2
JL^
_L*!Y_I: u = — .0000010
2048 v 3^ a+ 2i^vW 5 ft/
•° 000002
^kv+tp+ya* ^)t?-^> +
Sum= 4.7490149
82 Bulletin of the Bureau of Standards ivoi.b,no.i
2 2
^ir a n 1 n.2 = 25600 tt 2
or M=o. 001 1998950 henry.
The sum of the next two terms in the series is equal to about one part in ten million. The value of the mutual inductance is there- fore given with great precision by this formula. If the inner radius had been relatively smaller, the convergence would haVe been more rapid. We have, however, carried the computation much further than would in practice be necessary. Calculating the same problem by Searle and Airey's formula we have 2^=30 2/= 5 A=§ a = 4. N = 300 x N = 200 2
Aa 2 2
4
X*=~ 33-°° L2= i4375o
X^= 236.5 L,= -0.7959
x*= ~ 37° 1 £,= -1.71
X = 4869
t Z = —0.72
8
+
AWL
— =
i
—^ 1. 001 1500
AWL,X
irir" aooo°344
Z
—r -^= -0^0000033
Sum= 1.0011811
*MW> = II9 8 48o 5.
.-. M= 0.001 1998957 henry.
The terms neglected are less than one part in ten million. The
value of the mutual inductance found is only six parts in ten million greater than that found by Rditi's formula, and for this problem the convergence of Searle and Airey's formula is the more satisfactory. Groler] Formulas for Mutual and Self-Inductance 83
The same problem by Russell's formula (46) (extended to include six terms in each, part of the formula) gives
^=0.00119989 henry.
Of the three formulas Searle and Airey's is for this case the most convergent, and Russell's the least convergent. If the ratio of the radii was still more nearly equal to unity, Searle and Airey's for- mula would still be satisfactory; the convergence of Roiti's formula would, however, become poorer. If in the above problem the length 2/ of the inner coil be increased without changing the radii, the quantities L2n in Searle and Airey's formula would become rapidly larger, and the convergence would become poorer. Roiti's formula also becomes less satisfactory as / is increased. For 2/= 24, however, Searle and Airey's formula will still give the correct result to about one part in 1 00000, but Roiti's formula in this case converges very slowly. On the other hand, if the radius of the inner coil were smaller in the latter case, Roiti's formula could be used, but Searle and Airey's would not converge rapidly enough. This is shown in the next example.
EXAMPLE 38. ROITI'S FORMULA COMPARED WITH SEARLE AND AIREY'S FORMULA. COILS OF NEARLY EQUAL LENGTH Length of outer solenoid = 30 cm u inner u = 24 a=2 A-= 5 7\ ^ = 10 ?z 2 = = 40 In Roiti's formula = 2I.628l08 P* -Pi 2d term = + 0.062447 3d u - O.OO3707 4th « + O.OO3682 5th u - O.OOO724 u 6th + O.OOO5OI 7th _ - O.OOOI7O (<
8th u + O.OOOI59
Sum = 2I.690296
\TT a n n =
z % 2
640073-
x %
.'
:M= 0.00137008 henry.
84 Bulletin of the B\ ureau of Standards {Vol 8, No. 1
Searle and Airey's formula
**= - 33 L =-2 141
X,= 236.5 L, = 4826.5
^6= -I370-3 Z = — 160036
6
X = 4869
8
Z = 5.019 xio
8
7
Ao= -i.57 I Xio
8
-Xio= 15746
Z = 4.483 xio
12
9
d = 2$o
2
1. 000000
2d term = — 0.028200
3d = -0.012742
«
4th " = — 0.004845
5th « = — 0.001409
6th " = — 0.000251
7th " = + 0.000037
Sum = 0.952590
2-irVN^ 230400077-
2
.•.^=0.00136999 henry.
In this case we see that the higher order terms in Rditi's formula
arrange themselves in pairs of nearly equal values with opposite signs. The convergence is, therefore, better than appears at first sight, and the terms here neglected do not amount to more than one part in a million in M. Searle and Airey's formula does not con- verge so rapidly, the eighth and still higher order terms being appreciable. If the length of the inner solenoid were made still greater the L coefficients would become even larger than they are here, and the convergence would become unsatisfactory.
EXAMPLE 39. ROUTS AND SEARLE AND AIREY'S FORMULAS IN THE CASE OF SHORT COILS Length of the outer solenoid = 5 cm " " inner " =2 a=2 A = 10 In Roiti's formula: ft = 10.111873 £2=10.594808 1
Rosa Grover, Formulas for Mutual and Self-Inductance 85
xl
which used in the formula give a more accurate value of P1 + P2 P2-P1, viz: Pa-7>i= 04829359 2d term = 0.0063160 3d " = —0.0001968 4th " = 0.0003286 5th " =—0.0000274 6th = 0.0000250 " 7th = — 0.0000035 " 8th " = 0.0000023 Sum = 0.4893801 . 4ttV= i6tt 2
M
.:-?L= 77-2798I
In Searle and Airey's formula X = 2.75 2 L = 2 2
X J£
k
64
A= I
4
1203 T _~ 15
A6
~ioT4 Ar 16
_ 9265 r -
A 71
^"ESpi ~ 64
^=106.25 Zl0= ~^
1. 0000000
1st term = 0.0088581
2d " = 0.0000270
3d = —0.0000025
"
4th = — 0.000000
"
Sum = 1.0088825
27r a
2 2
N JV _
1 2
2
807T /2 1 ^ 2
d d
M
.;.-—= 77.27980
nt n2
The neglected terms in Searle and Airey's forumula are entirely inappreciable. The convergence of Roiti's formula is not quite so — j
86 Bulletin of the Bureau of Standards [Vol 8, No. i
good. The sum of the next two terms is such as to reduce M by three units in the last place, but the following terms do not decrease very rapidly. We may evidently regard the use of either formula as entirely justified in the problem. If the radius of the outer coil had been only one-half as great, the lengths of the two coils and the radius of the inner remaining unchanged, the value found by Roiti's formula would be in error by more than one part in ten thousand; the convergence of Searle and Airey's would, however, be satisfactory in this case. This formula would, on the other hand, be less convergent when the length of the inner coil is nearly as great as that of the outer coil. In general, it will be found that these two formulas between them cover sufficiently well the cases which may arise in practice.
EXAMPLE 40. GRAY'S FORMULA (41) COMPARED WITH ROUTS (39)
Let the winding be 20 turns per cm on each coil (Fig. 24),
«1 = «s 20.
A = 25 cm
N N = yi n Aa
X 2 x %
# = 10 cm
1
1 /• j?+aC4jj 1r»-A/7 /
M zMW
m
r* 1
1
/d \ir a «!«
U=25 d
i a=i0 / |
^=.0179057 henry.
—
1
1
1
We have also calculated the mutual
inductance for these coils by Roiti's
1
1
1
formula (39).
!
1
1
The value is, ^=.0179058, which
1
1
1
is practically identical with the value
by Gray's formula.
1
When A = 25 cm and a = 10 cm, N = 2oA^t> = 866.025 and JV =t 2
2oa-y[J= 346.4. As there must be an integral number of turns, let us suppose the winding is exactly 20 turns per cm on each coil and the lengths therefore 43.3 cm and 17.3 cm, respectively. Then d= -/r + A = -W 625 + ( ^-^ ) = 33-0715 cm. 2 % This does not exactly :
Grover ] Formulas for Mutual and Self-Inductauce 87
conform to the condition imposed in deriving the simple formula
(41) of Gray used above. Hence (41) will not be as exact with
these slightly altered dimensions, and we must calculate at least
one correction term to get an accurate value of M.
2
27T IOO X 866 X 346
-r.r^
By Gray's 1
formula
,/
«•
(41),
}
nsr
M=f* = .0178842
9
33.0715 Xio
'
on -
henry.
J
^
The first correction term in (43) increases this value to .0178854 henry. We will now calculate the mutual inductance of these coils by Roiti's formula (39)
^ = 25 2^ = 43.3 4 = 13.0 cm = 28.I78OO
Pi
a = 10 2/ = 17.3 4 = 30.3 cm p2
= 39.28218
P2-~Pl
= II.IO418
2nd term = + .22030
= — .OI78I
M=A ^ ni n,x n.32596
IO
h
3rd
4th
5th
= + .OI952
= + .OOI56
= .0178853 henry. 6th = — •00453
7th = + .00274
Sum = 11.32596
This differs from the result by Gray's formula by only 1 part in 178000.
EXAMPLE 41. GRAY'S FORMULA (40) COMPARED WITH NAGAOKA'S FORMULA (45) We will next consider a practical problem suggested by Prof. Nasmyth. 2x= 20.55 A = 6.44 N = 15 turns. x
2/= 27.38 = 4435 ^2 = 75 "
The distance between the adjacent ends of the two solenoids was 7.2cm. From this we find n x = 0.7296 turns per cm k x K = 0.042937 x
;z = 2.737
2
" " " /£
3
A"3 = 0.018274
^ = 20.89 £5^5 = 0.005193
^2 = 41.44 /k A" = 0.001423
7 7
^A^ = 0.000116
Sum = 0.067943
M = 1092.3 cm.
.-.
88 Bulletin of the Bureau of Standards \yoi.s,No.x
It is evident that the convergence is not rapid enough to give a
very precise value for the mutual inductance. We next divide the longer coil S into two sections C and D, such that C has $7 turns and D has 38 turns, C being the section nearer the other coil R. The axial lengths of these sections are, respectively, 13.51 and 13.87 cm. It would be just as well, if not better, to divide the coil into two equal sections of 37I/2 turns each. The division chosen was for greater convenience in the solution of the same problem by the method of quadratures. (Example 42.) We now proceed to find MRC and MRD . The L coefficients are much smaller than before on account of the ratio - being now a smaller than was previously the case, and the convergence is much more satisfactory. These coefficients would be still smaller if we had divided coil R instead of S into two sections, measuring the x*s from the center of R instead of using S for the reference coil as is done here. This advantage would, nevertheless, be in large measure offset by the smaller values of the distances rx and r2 .
We find for MRC
K
k x x = 0.048894
k3 jYz = 0.006520
K
k h b = 0.000051
Sum = 0.055465
.'.MRC = 891.7 cm
and for M RD ^^ = 0.011549 £3^3 = 0.000613 k K = 0.000004 b b
Sum = 0.012166
.\44d= 195.6 cm.
Consequently M=M M RC + RJ) = 1087.3 cm -
To test the correctness of this value, the coil R was divided into two sections A and B (B being the section nearer to S), and the four mutual inductances between these sections and the two sections C and D of the coil S were calculated. 1
Grfver] Formulas for Mutual and Self-Inductance ' 89
MXG Mj>Q M^ ^BD
£^ = 0.012024 0.036869 0.003893 O.OO7657
k K = 0.000863
z % 0.005654 0.000141 O.OOO47I
/^ A" = 0.000004 0.000047 0.00000 O.OOOOO4
5 5
M =2oy.2 M = 684.4 ^ =
0.012891 0.042570 0.004035 0.008132
AC BC 64.9 M =
BT) 130.7
ikf=sum = 1087.2 cm.
The only component for which, the convergence was not entirely satisfactory was M Here the sections are relatively near together BC .
and the coefficients L and are not very favorable. X Accordingly M BQ was calculated by two other methods (a) by dividing B into two sections, H and J, and by calculating AfK0 and AfJC (b) by M ,
dividing C into two sections, F and G, and by calculating BF and M B(i . The first procedure, on aceount of the relatively smaller values of r and r2 did not give a satisfactory degree of convergence. x , The latter, however, is better, the values found being
M = BC 463.8 + 220.0 = 683.8
Using this value instead of the above the value of M is 1086.6 cm. As the final check we have calculated the mutual inductance by Nagaoka's formula (45). The entire calculation has to be carried through for four different values of *:, viz, 55.13, 34.58, 27.75, an<^ 7.20. The corresponding values of I are I = 60.041802 x
f = 38.047638
2
73 = 30.811676
A =10.333503
and (/, 4- /4 ) - (/8 + /3 ) - 7o«3753°5 - 68.859314 = i-5 I 599 I .-. M= 47m 1 ;z 2 ^#(i.5i599i) = 1086.55 cm.
An inspection of the various details of the calculation shows that
the last figure may be in error by several units, although the utmost precision of which the seven-place logarithms are capable was striven : 1
90 Bulletin of the Bureau of Standards [Voi.8,No.i
for. Of course by carrying the computation of the various quanti- ties to a still greater number of decimal places, the accuracy of the result would be enhanced. Similarly the component values of the mutual inductance by Gray's formula would have been more accu- rate if we had not stopped with k b b K Since the dimensions of .
such coils would not ordinarily be obtained with greater precision than the accuracy here attained by Gray's formula, it is evident that the latter is for such cases much to be preferred to Nagaoka's for- mula, and the same would be true if the number of components were considerably increased. Nagaoka's formula has nevertheless the advantage in checking other formulas.
EXAMPLE 42. FORMULA OF QUADRATURES
The problem treated in the preceding example may also be solved by the formula of quadratures, using formula (8) for the calculation of the mutual inductance of the various pairs of circles. In general the method is not so accurate as that in the preceding example, and no time is saved. Only the results are here given, together with those by Gray's formula for comparison. Single coils Two sections Four sections ^=964.1 M = M = ^rd= RC
M= 1041.8
848.1
193-7 ^
^AC= 2057
= 669.5
875.2
BF
M =
BG
M =
504.5
182.2
686.7
-
BC
^AD= 66.5
^BD=I3°4
^RD = I96.9
M= I072.
Using the value of MBC in the last column, M= 1089.3. By Gray's formula Single coils Two sections Four sections M= 1092.3 M = RC 891.7 J/AC = 207.2 M = CK 463.8 J/RD = 195.6 M = BC 684.4 M= CJ 220.0 M= 1087.3 891.6 M = BC 683.8 M = AB 64.9 M = BD 130.7 ^RD = 195-6 M= 1087.2 Using the value of MBC in the last column, M= 1086.6. GroveA Formulas for Mutual and Self-Inductance 91
EXAMPLE 43. MUTUAL INDUCTANCE OF CONCENTRIC COAXIAL SOLENOIDS BY NAGAOKA'S FORMULA (45) 2x = 200 cm 2/= 20 A- 15 a= 10 51 This pair of coils was used by Cohen in testing his absolute formula. He gave as the result M=\irn x n % (6213.4). The same problem was worked out by Nagaoka 52 as an illustration of the use of his formula, the value of M found by him being M = dt irn1 n % (6213.51). We have repeated his calculation, which was given only in abbreviated form, and agree substantially with his result, the value found being M=^ irn 1 n 2 (6213.52). r Using seven-place loga- rithms it is very difficult to be sure of the last place of decimals given here. On the other hand, we find with Roiti's formula, only three terms being necessary Af==4.7rn 1 n 2 (6213.509), and the same number of terms in Searle and Airey's formula give M= 4'7rn 1 7t 2
(6213.510) with no uncertainty greater than one unit in the last place given. Olshausen found for the same coils the values \irn x n % (6213.77), and 473-72^2(6213.63), using two methods of calculation
in his formulas (21) and (61) and with m= —— (A - 2 4- a) 4- c* -4 . By the Kirchhoff formula he found 4^^^(6212.9). This is, however, an unfavorable case for this formula, since the angles <p and 6, on which the incomplete integrals E (<p, k ) and F(q>, k') depend, are too near r
90 ° to allow of accurate interpolation in Iyegendre's tables. These differing results by the various absolute formulas, which arise from the fact, that in all of them the auxiliary quantities must be calculated with a considerably greater degree of accuracy than that desired in the result, serve to emphasize the advantage of the series formulas. In the great majority of practical cases the values, found by the use of series formulas, are not only obtained with a much smaller expenditure of labor, but are more accurate than when an absolute formula is used. 51 This Bulletin, 3, p. 8; 1907. 52 Jour. Tokyo, Math. Phys. Soc. (2), 4, p. 284; 1908. =
92 Bulletin of the Bureau of Standards [ Vol. 8, No. i
We reproduce below the principal results in the calculation of
this problem by Nagaoka's formula.
A + a = 2S A-a = s A' a*=i25
(A + af = 625 (A-af = 2 5
The argument of It = — 1 is c= no; that of I. = — It is e=go.
3 2
C=110 C=90
log a = I-374979 6 I-3749796
log 2Aaa = 1. 8521009 1. 852IOO9
P = V^-e = — 170.09225
1 1
-II3-86339
P = $7j-e = + 0.3514302
2 2 + O.3514302
P = V?'-es = + 8.7857551
3 + 8.785755I
z> = ±(P + P + P = - 53- 6 5 I 687
1 2 3) - 34.908733
^2 = ^3 84343249 84343249
<?
x
^ = 3 178.87801 I22.649I6
» ^1 "~ ^2
= 170.44368 II4.2I483
0.9761398 O.965OO37
V^l-^3
1 + v# 1.9879979 1. 9823461
log£ = 3
2.6734934 2.8373857
? = 0.0030186570 O.OO44528OIO
log &>! (1st equat.) = 1.0750696 1. 1594886
" (2d « ) = T.0750696 1. 1 594886
" (3d " )
= T.0750696 T.i 594887
^2+ ^3 = 9-!37i853 9^37^53
/>!+ 2/> = -169.38939 -113.16053
^1+ 2P = -152.52074 3
— 96.29188
/'.)- -
f« + 777.08214 -520.19542
-^V + 2P = -
(
1 8) 238.11479 -159.07205
— Sum = + 1015.19693 679.26747
X©i=^= 120.67564 98.068477
pz>03 "(o)_
26.73272 21.064838
4®i ^3(0)
Diff.^/7 - 93.94292 77.003639
(other equat.) = 93.942921 77.003659
Grfver] Formulas for Mutual and Self-Inductance 93
s = 0.33164947 - O.33084473
s
109.86656 74.300356
q •
1 + 2^* cos 47TW = 1.0000042 1.OOOO087
b= 109.86702 74.301000
\W == 0.99093909 O.98663068
4^-1 [i-^ (^-i)] 2 0.00399641 O.OO587502 Sum = 0.99493550 °-9925057° /" = - 35-8i5io7 — 29.231710
l=I>+I" = 58.127814 47-77I939
2
/x-/2 = 2(58.127814-47.77 x 939)
= 2(10.355875)
i^r= ^nrn x 7t % Aa. 2(1\ — Z>)
=
4^^(6213.52)
In the calculation of / we have used the second value found for /' with c=no and the mean of the two values for I' with c= 90. EXAMPLE 44. SHORT SECONDARY ON THE OUTSIDE OF A LONG PRIMARY Length of primary = 200 cm " " secondary =5 Radius of primary =4=a " " secondary = 5 = A
n x —\o n 2 = /\.o
In Roiti's formula: = 97.62811 /9 2
p = 102.62188
2
^~^ = = 4.9937586
;TXT
P1+ P2
AV/ 1 1 \
°- QQOO° 75
-si?-?)"
Sum = 4.9937661
a
47r # « 1 # a
8
= 25 6oo7r
2
.*. M= 0.001 261 7342 henry.
21674 — 12
7
: 1 :
94 Bulletin of the Bureau of Standards [Voi.8,No.z
In Searle and Airey's formula:
d = 1 002 5
2
-A^ iV2 = 400000
= 20 -=g
^ # 8
^=-1597 A= 1-4375
A = —0.796
1. 00000000
1st term = 0.00000071
2d " =— 1.3X10- 10
Sum = 1. 0000007
.*. M— 0.00 1 261 7342 henry
in exact agreement with the above. Both formulas are very rapidly convergent, and give as nearly the same value for M as can be cal- culated with seven-place logarithms.
EXAMPLE 45. COILS OF EQUAL RADII NEAR TOGETHER, BY
FORMULA (48)
Lengths of the coils 4 cm and 6 cm = b x and b 2
Radius " " " 20 cm = a
Distance between centers 10 cm = d
The calculation of the quantities in the parentheses is as follows
First Second Third Fourth
1st term = 0.088055 O.OI562 O.OO47 -0022
2d term = —0.012694 O.OOI26 — 0.0002
1
3d term = 0.001232 O.OOI30
4th term = —0.000104
Sum = 0.076489 O.Ol8l8 0.0045
1
.0022
The expression for JM then gives
First term =0.0233240
Second " =0.0011047
Third " =0.0000973
Fourth " =0.0000113
Sum = 0.0245373
Gaoler] Formulas for Mutual and Self-Inductance 95
The value of M calculated for two circles of radius 20 cm and at a distance apart = 10 cm was found by (4) and checked by (1) to be M N N = 41m (0.8853877) X Z
4M«
= 47m (0.0245373)
M
N N = ^7ra (0.9099250)
X %
This was checked by means of (51) with the result, (assuming one turn per cm of the length of the coils) ^abc =4t* (430-33973 6) Lc = \ira (74.324564) Sum =4.7ra (504.664300)
ZAC =4^(194.210135)
ZbC = 471-0 (266.777705)
Sum =\ira (460.987840)
— Dim = \ira (21.838230)
2
Dividing by 24, the product of the number of turns assumed in calculating the self-inductances,
-yr-^ = 47m (0.9099264)
which agrees with the value by (48) to about one and a half in a million. For these coils, therefore, (48) is adequate to give a high degree of precision. If the distance between the same coils were, however,
smaller, or if the lengths of the coils were greater the accuracy would not be so great, and it might be necessary to use (51). The latter, should, however, not be used when (48) converges well, since
to get the same accuracy the calculation of the four self-inductances must be carried out to a greater number of decimal places than appear in the value of M. For the rapid convergence of (48) the
ratios -* -f and - should all be small. d d a For the more unfavorable case bx = 6, b % = 10, d= 10, a = 20 the value of JM comes out too small by three parts in ten thousand. 96 Bulletin of the Bureau of Standards [Voi.8,No.z
EXAMPLE 46. ROSA-WEINSTEIN FORMULA (50). FOR COILS FARTHER APART As a rather unfavorable case we may take ^=10 &j=2o d=$o a = 25
£= sin 7 = .
° = —= = cos 7
V5000 V2
2 2
cos 7 _ sin 7 _ 1
24^ 24^ 120000
ax = 500 a = x 500
-or3 =- 4.55 2a 3 = 9.10
-3^3 cos 2 7= - 6.825 +3^3 cos
2
7=+ 6.825
8a3 cos4 7= + 9.10 8a3 cos 7=4
9.10
Sum = 497-725 Sum = 525.025
P= .0041477 = .0043752
F= 1.854075
E= 1.350644
F-E= 0.503431
(F - E)P+ EQ = 0.0079974
.-.JM= 1.7766
Terms in the 6th and 8th differentials =0.0016
Sum = 1.7782
From formula (19) which applies to the two circles at the centers of these coils M = 8599 X 25 = 354 5° i-4 I 6
M = 35.4650+1.7782 = 37.2432
N,N %
If we calculate the mutual inductance by formula (51) we find
^abc = 3792.226i LAC = 2350.4870
Zc = 1667.7268 Zbc = 3062.0405
5459.9529 5412.5275
- Diff. = 23.7127
Dividing by 200 = o. 1 1 85635
M
Q
AW = 37.2478
:
Grwer] Formulas for Mutual and Self-Inductance 97
which is more than one in ten thousand greater than the value by (50). If the coils had been shorter and their diameter had been
greater than the distance between their medial planes, the quantities P and Q in (50) would have been more convergent and the value of JM would have been more nearly correct. The accuracy here obtained would, however, suffice in many cases. This formula when applied to the coils in the preceding problem
gives a very accurate result viz, = 477^(0.909932), or about six in a million too large. (The terms in the sixth and eight order differentials as calculated by (48) are taken into account in this result.) The mutual inductance of the coils in this example could also be calculated with a good degree of accuracy by Gray's formula.
EXAMPLE 47. METHOD OF OBTAINING THE DIMENSIONS OF THE
EQUIVALENT CURRENT SHEETS
Suppose it is desired to obtain the mutual inductance of two solenoids, whose measured dimensions are as follows Coil I is wound with 100 turns of insulated wire of 0.15 cm covered diameter, the successive turns being in contact. The measured external diameter of the coil is 50.4 cm. Coil II is wound with 50 turns of bare wire, o. 1 cm in diameter, in a thread of 2 mm pitch. The diameter measured over the wire is 10.25 cm '
Then the mean radius of coil I, to the center of the wire, is equal
to - (50.4 — 0.15) or 25.125 cm. The length of the equivalent cur- rent sheet will be the distance between the center of the first and the one hundred and first wire, or one hundred times the covered diameter of the wire; that is, 15 cm. Since the turns are in contact, the equivalent length may, in this case, also be found by measuring the over-all length of the winding, including the insulation. Both these methods are equivalent to taking one hundred times the pitch of the winding, which, in this case, is equal to the covered diameter of the wire. 98 Bulletin of the Bureau of Standards [Voi.8,No.i
For coil II, the equivaleut radius is - (10.25 — 0.1) = 5.075 cm -
The equivalent length is fifty times 0.2 = 10 cm. The dimensions found in this way for coils I and II are to be used in the appropriate current sheet formula. (See p. 73.)
THE MUTUAL INDUCTANCE OF A CIRCLE AND A COAXIAL SINGLE-LAYER COIL LORENZ'S FORMULA The problem of finding the mutual inductance of a circle and a 53 coaxial single layer winding was first solved by Lorenz. Assuming that the current was uniformly distributed over the surface of the cylinder, forming a current sheet, he integrated over the length of the cylinder the expression for the mutual inductance of a circular element and the given circle. This expression is an elliptic inte- gral. Lorenz reduced the integrated form to a series and gave the following formula for the mutual inductance of the disk and solenoid of what is now called the Lorenz apparatus. He called it, however, the constant of the apparatus instead of mutual inductance, and denoted it by C. It is of course the whole number of lines of magnetic force passing through the disk due to unit current in the surrounding solenoid.
0(«)=-^[ i+ §£ + ^(H [53]
p = radius of the disk, Fig. 25. r= radius of the solenoid. 2x = length of winding of solenoid. a = p/r = ratio of the two radii.
d= —n = distance between 2,X centers of successive turns of wire.
x* + r
2
2
r
63
Wied. Annalen, 25, p. i; 1885. Oeuvres Scientifiques, 2, p. 162.
: 1
tower] Formulas for Mutual and Self-Inductance 99
If the disk be not exactly in the mean plane of the solenoid, and
x be the distance from the plane of the disk to one end of the x
solenoid and x to the other, ^ % 2x- x? + r 2 x* + r2 f
Then Q(a ) is found by substi- x | r j
tuting the values of a x in equa- L tion (53) above, and Q(a2 ) by using the value of a 2 for a in — L the same equation. The sum of these two quantities multiplied
by —^— gives the constant of the instrument: that is, the mutual " inductance sought. As Lorenz gave the expression Fl £ 25 for the general term of (53), his equation can be extended. The following is the general term
yW-2*2_f 2.4 2m 1.2 (m + i)da m\ a ) JONES'S FORMULAS 54 Two solutions of the above problem were given by Jones, both , ^ ^ in terms of elliptic integrals. The cur- -A rent was considered to flow not in a ^ current sheet, but along a spiral wind- ing or helix. The first solution was in the form of a series, convergent only when O A, x Fig. 26, is less than the difference in the radii of inner and outer coils; that is, when O A is less than A — a. x
As this is a serious limitation, and the
formula is a laborious one to use, it is not
here given. The second solution is
exact and general, and is in terms of
elliptic integrals of all three kinds.
Fi s- 26 The second formula is as follows:
M = ®{A + a)ck\^ + "(F-jA
e
c^
C
c* [54 ]
54
J. V. Jones, Proc. Roy. Soc, 63, p. 198; 1898. Also, Trans. Roy. Soc, 182, A;
Jones's first formula was given in Phil. Mag., 27, p. 61; 1889.
ioo Bulletin of the Bureau of Standards [Voi.8,No.i
M = mutual iuductance
e of helix O A, Fig. 26, with respect
x
to the disk S iu the plane of one end.
<£) = 27772, i/n = pitch of winding, & = whole angle of winding.
F E, and II are the complete elliptic integrals to modulus k
} )
where
II, the complete elliptic integral of the third kind, can be expressed
in terms of incomplete integrals of the first and second kinds, and the value of M can then be 9 calculated by the help of Legendre's tables. (See example 50.) The calculation is, however, extremely tedious, especially when the value is to be determined with high precision. Campbell has given Jones's formula (54) a slightly different form, 55 somewhat more convenient in calculation, as follows:
IT- 2 WHlnXA+jjfeF-£)+d=5+ [55]
j
where n x is the same as n above, the number of turns per cm on the solenoid, n % is the number of turns in the secondary coil (in the above case it was taken as one), \ A is the greater and a the less J I
\ , of the two radii (in the above
_j jx- £ P case A was the radius of the \ solenoid and a of the circle within), and
^ = F(k)E(k'^)-
Axis \_F{k)-E{k)-\h{k'^)-^
Fig. 27
where F[k) and E{k) are the
complete elliptic integrals to modulus k, and J*(k',/3) and E{k' ,/3) are the incomplete elliptic integrals to modulus k' and amplitude /3; k' = COS7, sin /3= c jk ; k, c, and c' are given above. f f If a second-
65
A. Campbell, Proc. Roy. Soc, A, 79, p. 428; 1907. There is a misprint in the
formula as given in Campbell's paper. It was, however, used correctly in the
numerical calculations given in the paper.
G°rove^
Formulas for Mutual and Self-Inductance 101
ary circle or coil has a radius greater than that of the solenoid, the same formula can be used if A is taken for the radius of the larger secondary and a is the radius of the solenoid (Fig. 27).
ROSA'S FORMULA 56
The following formula gives the mutual inductance of a single
layer coil of length x and a coaxial circle of radius a in the plane of one end of the coil, as shown in Fig. 26. It is the same quan-
tity represented by M of equations (53) and (55) and M e of (54).
2 2
NV $a A 2 2 A*
n , 2ir a . 5 a'A* v 35 a* A' 63 a*
+
d™
X + 8l 2~d ^ X" + 2 ' L56J
4o 9 6
* '
9 J
2
=3~4
Z"2
Xv^t 10A>
x*
+
Sif
^
Y 00 * 35_35^! 00 , ^T ^ „* 16 2 A 2 '
"~A* ^A (
4A^^-^ ^ x ,x\
2
61 105 , x? x 8
X = j2 T_ * ,
+
^ ~i28 10 32 A ^ 2 8 A' 1D ^A 6 ^ 55 A S 4 ^ 10
(For general coefficient, see p. 63.)
a = radius of disk or circle S, Fig. 26.
A = radius of the solenoid.
x = length O A of one end of the solenoid.
x
d= Jx + A = hali the diagonal of the solenoid.
}
i 2
N
is the whole number of turns of wire in the length x.
This formula is very easy to use in numerical calculation, not- a 2A 2 withstanding it looks somewhat elaborate. The logarithm of — ^-,
multiplied by 2, 3, 4, etc., gives the logarithm of the corresponding factor in each of the other terms. Similarly, the various terms x 2
X X±,2y etc., contain only powers of —^ A besides the numerical
56
This Bulletin, 3, p. 209; 1907.
: i
102 Bulletin of the Bureau of Standards. [Vol.8, No.
coefficients. It is hence a far simpler matter to compute with M high precision by this formula than by Jones's formula, the latter containing as it does elliptic integrals of all three kinds and involv- ing the tedius interpolations for incomplete elliptic integrals. If the secondary circle has a larger radius than the solenoid, A will be the radius of the circleand a the radius of solenoid. In every case A is the greater and a the less of the two radii, and d is +*r Equation (56) may be written
M= ~~da n xS
27r x
where n x is the number of turns of wire per cm, x is the length of the coil, Fig. 26, and 6* is the value of the quantity in brackets in (56), which is always somewhat greater than unity. This may also be put as follows M= a n/ ~^\s= a n RS 2 277 2 x
or, [57] M=a n K % x
The quantity R depends on x\d; that is, only upon the shape of the solenoid. 5* depends upon xjA ,
a/Ay and Ajd; that is, upon the
relative sizes of the inner circle
and the solenoid and the shape
of the solenoid. If we have the
value of RS, or K oi equation
(57) for a given solenoid and
circle, we can get M by multi-
plying by a 2 n x1 and for any other
system of similar shape but dif-
ferent size by multiplying the
same value of K by a*n x
. The
values of the constant K for
Fig. 28 various values of a\A and xjA
are given in Table III, page 193.
be in the center of a solenoid of length 2x
If the disk or circle
(Fig. 28), the value of M is of course double that given by using x.
If it be not quite in the center, the value of M must be calculated for each end separately. Rosa "I Formulas for Mutual a?id Self-Inductance Grover\ 103
For illustrations and tests of the above formulas, see examples 48 to 51, pages 1 03-1 10.
EXAMPLES ILLUSTRATING THE FORMULAS FOR THE MUTUAL INDUC- TANCE OF A CIRCLE AND A COAXIAL SOLENOID EXAMPLE 48. ROSA'S FORMULA (56) COMPARED WITH JONES'S FORMULA (54) Professor Jones gave the calculations by formula (54) of the con- stant of the Iyorenz apparatus made for McGill University, obtaining the values given below, the second value being that obtained after the plate had been reground and again measured. A calculation 57 of the same two cases by formula (56) gives very closely agreeing results.
ist value 2nd value, disk slightly smaller
By formula (54) M= 18,056.36 18,042.52
" " (56)^=18,056.46 18,042.74
Difference —.10 —.22
These differences, amounting to five parts in a million in the first case and twelve parts in a million in the second case, are wholly negli- gible in the most refined experimental work.
EXAMPLE 49. FORMULA (56) COMPARED WITH JONES'S FIRST FORMULA Take as a second example the case given by Jones 58 to illustrate his first formula.
A = 10 inches a = 5 inches x = 2 inches
=
,a aA 2 2
2500 A = I-363S733 2 2
log ,
10816 d
ist term = 1.0000000
2 " = .0866771
^ = 2.8400 3 " = .OH8537
X = 2.1064
i 4 " = .0017781
X = 1.5308
6 5 " = .0002670
= 1.0173
-x; 6 " = .0000379
X = 0.5815
10 7 " = .0000046
27rV Sum= 1.1006184
48.38972
d
57 This Bulletin, 3, p. 218; 1907. 58 Phil. Mag., 27, p. 61; 1889. In this example, P should be 0.654870 instead of 0.54870, as printed in Jones's article. io4 Bulletin of the Bureau of Standards [ Vol. 8, No. i
.•.^=53.25861 iV, N being the number of turns of wire on the coil.
Jones gives M= 53.25879 N.
The difference between these values is three parts in a million. EXAMPLE 50. CALCULATION OF CONSTANT OF AYRTON-JONES CURRENT BALANCE BY FORMULAS (54) AND (56)
As a further test of the formulas let us calculate the constant of an electro-dynamometer or current balance of the Ayrton-Jones type, 69 of which AB, Fig. 29, is the upper fixed coil and ED is the moving coil, the circle S at the upper end lying in the plane through the middle of AB and the circle R at the lower end of ED lying in the middle plane of the lower fixed coil BC.
•
^*\
U-]
j
\
\
\ a
1
:
\
\ Pt
•
:
i
B |
\
\ J
;
: \
j ] \ !
o\
1
\a .; ;
£
1
n B
:
I 4 a, >
I >— —— —— Ar— — -*
0&1
Fig. 29
Assume the dimensions as follows: A = 16 cm = radius of fixed coil, Fig. 29. 10 cm = radius of moving coil. 8 cm = half length of AB = O A x
x = 24 cm = 1.5 times AB = A
9 2
10 = number of turns per cm
80 = number of turns in distance 1
A=.ar
1,
Fig. 29.
240 = number of turns in distance 2
A=x 2
This Bulletin, 3, p. 226; 1907.
3
Rosa Grover] Formulas for Mutual and Self-Inductance 105
d = -JA + x\ = 8^ = diagonal AP
±
2
X,
Fig. 29.
d = -J A + x\ = 8 V13 = diagonal AP
2
2
2
We have to determine two mntnal inductances, namely, Af be- s
tween the coil OjA of 80 turns on the circle S, and R between the M coil 2 A of 240 turns on the circle R. In each case the circle is in the plane passing through the lower end of the coil. N Formula (56) will be used, taking ly x„ and dx in the first case and N^ x2) and d2 in the second case.
For M s For Jf/R
A 16 cm 16 cm a 10 10 X 8 24 A 2 256 256 X* 64 576 JV=nx 80 240 2 d 320 832 log d 2 2.5051500 2.9201233 , a2 A 1-3979400 2.5679934
x2
i-39794oo 0.1 76091
x, + 2.000 — 6.00 x, + 0.250 + 0.25 X. -o-9375 + 23-5 x> -1.203 -45-7 •^10 -0.562 - 49.0 1st term 1. 0000000 1. 0000000 2d " + -0937500 .0138683 3d « + .0097656 .0006411 u 4th + .0002670 .0000009 5th " — .0002253 .0000027 6th " — .0000662 .0000002 7th " — .0000072 .0000000 Sum = >S 1. 1034839 1.0132306 i
io6 Bulletin of the Bureau of Standards [Vol.8, No.
logS, = 0.0427660 log S% O.OO57083 " 27T2 = I.2953298 " 27T 2 I.2953298 " a\ == 100) = 2.0000000 " # 2 (=ioo) = 2.0000000 " ^i( = = 80) = 1.9030900 " A = 24o) r 2( = 2.38o2II2 5.2411858 5.6812493 " dx = 1^525750 " d2 = 1. 4600616
log M B = 3.9886108 log M R = 4.22II877 •:MS = 9741.16 M R = 16641.32 THE SAME EXAMPLE BY JONES'S FORMULA We will now calculate M and MR by Jones's second formula given s
above, using also the following equation to find F— II: — *w"-) £' 2 sin/3cOS/3(^-II) M w,, Q , , zr/Mzr^/ OS T?U,\I?{lt j. y^ }J yn,. ox }Mj *"
c
ForJ/s For M R
A 16 cm 16 cm a 10 10 X 8 24 %=2irN l6o 7T 480 7T 2-4 Aa O.973OO85 O.973OO85 A+a O.23O7692 O.2307692 2^Aa O.9299812 O.71497OI J(A+af+x> O.3676073 O.699155O
log sin fi 1 sin yS = - 9-7977938 9.5l86043
F{k) 2.4373371 I.8636661 E{k) I- 1323456 1.3449927 F-E I.5088957 I. OI46546
F{k\ /3) O.6852557 0.3394833 E{k\ 0) O.672I988 0.3333201 k n sin /3 cos /3 {F--n) -O.8266738 -I. I256799 c
-O.685I799 — O.4045298
GrTver] Formulas for Mutual and Self-Inductance 107
^g |^^+^(F-n)J 1.9157773 1.7854187
log (%(A + a)ck) 4.0728340 4.4357689 log M 3.98861 13 4.2211876 M = 9741.17 cm s M&— 16641.33 J/s differs from trie value obtained by formula (16) by one part in a million,M R is identical. M s is the mutual inductance of the winding O A on S. The x
inductance M x of the whole coil AB on S is twice as much, that is ^=19482.34 The inductance of AB on R is MR above, minus the inductance of 2 B on R which is the same as that of OjA on S, that is, M s.
Therefore,
M = 16641.32 — 9741.17 = 6900.15
%
Hence M — M = 12582.19 cm.
1 2
The force of attraction of the one winding AB in dynes is
The force due to the second winding BC is equal to this. Sup-
pose i1 = %=i ampere = 0.1 c.g.s. unit of current and ;z 3 =io turns per cm. Then ^4^2 = 0.10
.•./"=o.2oX 12582.19 dynes = 2516.438 dynes 2/= 5032.876 dynes = change of force on reversal of current = 5.1356 gms where g = 980. If there are two sets of one on each side of the balance, as coils,
in the ampere balance built for the National Physical Laboratory, the force would be doubled again. In calculating the mutual inductance of the disk and surrounding solenoid in the Lorenz apparatus the series (56) will be more con- vergent when the winding is long, and of course more convergent when the disk is not of too great diameter. io8 Bulletin of the Bureau of Standards I Vol 8, No. I
EXAMPLE 51. MUTUAL INDUCTANCE OF CAMPBELL'S FORM OF STAND- ARD BY FORMULAS (55) AND (56) A cylinder 20 cm in diameter has two coils of 50 turns each wound as shown in Fig. 30, each covering 5 cm ( = AB) with an interval of 10 cm between ( = AA'). A secondary coil of 1000 turns of finer wire is wound in a channel S, with a mean radius of 14.5 cm. The magnetic field near S, due to the double solenoid, is very weak, and is zero at some point; at this place M will be a maximum, and variations in Mdue to small changes in A will be very small. To calculate M for the solenoid AB and the coil S, we take two cases, as in the preceding example. First, M for S and a winding
2 B of 100 turns; second, M 'for S and 2 A of 50 turns. The differ- ence will be M for S and the actual winding AB. Or, supposing
A*— 5 crrrr-^
a=l0
OOOOOOO OOOOOOO
Fig. 30
AB to have 100 turns, M will be the same as for AB of 50 and A' B' of 50. Using formula (55) we have the following values: :
Grovcr] Formulas for Mutual and Self-Inductance 109
For M
x
For M
2
a= IO = 10
A= 14-5 = i4-5
x=b = IO = 5-o
log k = 1.9590874 = 1. 98366715
7= = ,,
65°3 lf 7"'32 74
/
2 3 38 .88
k' = V0.1717243 = V0.0723711
P= 26°i8 36 ,, .85 = 43°3 33 /, -o6
/ /
7' = 24°28 52 // .68 = i5°36 2i // .i2
, /
F= 2.3267801 = 2.7312000
E= 1.1590043 =- 1.0812388
. $?-*>- 1.2613045 = 1.6839704
H&',®= 0.4618972 = 0.7561693
E(k'fi) = o-45 6 53*4 = 0.7469284
+ «-- 1.0479404 = -o-7784352
-
A-a ,
= -0.7005918
-04715732
^
•
_V-*) + = 0.7897313 = 0.9833786
nx n2 = 200,000 = 100,000
^1= 24,313,940 cm 2
=M 15,137,940 cm
= 24.31394 millihenrys = 15.13794 milli-
henrys
M= M — M = g.iy6o 1 2
millihenrys.
Campbell gives 60 the value of M as 9.1762 millihenrys, but for want of any particulars of his calculation we do not know wherein the difference lies. We have worked this problem out also by formula (56) with the following results M = 24.31387 millihenrys x
^= 15.13857 « «
'
M = 9.17530 " "
The value of M agrees with that found by (55) to about two parts M x
in a million. M however, a little larger, making 2 is, smaller. This is due to the fact that formula (56) is not as convergent for 60 A. Campbell, Proc. Roy. Soc, 79, p. 428; 1907. 21674 — 12 8 no Bulletin of the Bureau of Standards \yoi.8,No.i
x=§ in this problem as for x= 10, and hence the terms neglected after the seventh are appreciable. Hence, for so short a coil as this, formula (54) or (55) will give a more accurate result than (56).
5. THE SELF-INDUCTANCE OF A CIRCULAR RING OF
CIRCULAR SECTION
KIRCHHOFF'S FORMULA
The formula for the self-inductance of a circle was first given by
Kirchhon061 in the following- '& form:
Z=2/jlogJ- I.508) [58]
where / is the circumference of the circular con-
£ ductor and p is the radius of its cross section. This
I
is equivalent to the following:
Z = 4ttJ log -£-1.751 [59]
1 p J
a being the radius of the circle, Fig. 31. These
formulas are approximate, being more nearly cor-
rect as the ratio p\a is smaller
O Fig. 31
MAXWELL'S FORMULA
^ more accurate expression, obtained by means
of Maxwell's principle of the geometrical mean distance, is the fol- lowing :
Substituting in this equation the value of the geometrical mean dis- tance for a circular area, R = pe~* = .7788/3, we obtain 62
L= 4™)/ + 0.1 37 ^\ log -^-.0095^-1.75
1 1 [61]
This is a very accurate formula for circles in which the radius of section p is' very small in comparison with the radius a of the circle. The geometrical mean distance has, however, been R computed on the supposition of a linear conductor, and can only
61
Pogg. Annalen, 121, p. 551; 1864.
62
Wied. Annalen, 53, p. 935; 1894.
1
Grwer] Formulas for Mutual and Self-Inductance 1 1
be applied to circles of relatively small value of p/a and the square )
of trie geometrical mean distance is used for the arithmetical mean square distance in the second order terms. We must therefore expect an appreciable error in formula (61) when the ratio pja is not very small. Formulas (58), (59), and (61) have been deduced on the supposition of a uniform distribution of the current over the cross section of the ring. If the ring is a hollow, circular, thin tube, or if the current in the ring is alternating and of extremely high frequency, so that it can be regarded as flowing on the surface of the ring, the geometrical mean distance for the section would be the radius /o, and we should have instead of (61) the following by substituting R = p,
In the case of solid rings carrying alternating currents of moder- ate frequency the value of L would be somewhere between the values given by (61) and (62).
RAYLEIGH AND NIVEN'S FORMULA
Rayleigh and Niven gave, 63 without proof, the following formula 64 for a circular coiloi n turns and of circular section, which is more nearly exact than either of the preceding:
7 + ^" 75
z=4
H(I+ £ 8
)
log I'
l
[63]
When »=2i, this will be the self-inductance of a single circular 65 ring. This formula neglects higher powers of £ than the second,
63 Rayleigh's Collected Papers, Vol. II, p. 15. 64 Neglecting the correction for effect of insulation and shape of section of the separate wires. 65 Max Wien, Wied. Annalen, 53, p. 928, 1894, derived by direct integration of Maxwell's formula (12) over the cross section of the ring, the formula
/-= 4TO
{(
I+
B^)
log 7^ oo83 |- 75
- I-
|
It was shown, however, by Terezawa, Tokyo Math. Phys. Soc, 5, p. 84, 1909, that this formula is in error, the correct result being identical with that of Rayleigh and Niven (63). This result was verified by Mr. Cohen at the Bureau of Standards in 1909, and quite recently independently by Mr. T. J. Bromwichof Cambridge, England. The error of Wien's expression is in practical cases of no importance. —
ii2 Bulletin of the Bureau of Standards [Voi.8,No.i
and its error therefore depends on the magnitude of the ratio of the radius of the cross section to the radius of the ring. Assuming, as
is probably justified, that the coefficients of the terms in ( —) , are of
the same magnitude, or smaller, than those of the terms in ( ^ ) , the
error will not be greater than even for ^ =0.1, an exception- iooooo a ally unfavorable case. If used for a coil of more than one turn, the expression for L must be corrected for the space occupied by the insulation between 66 the wires and for the shape of the section.
SELF-INDUCTANCE OF A TUBE BENT INTO A CIRCLE
Suppose that the cross section of the ring is not solid, but is an annulus bounded by two concentric circles of radii px and p 2 p 2 being ,
the larger. Then assuming the current to be uniformly distributed 67 over the cross section, we find by means of Wien's method
^44( I+ ^->og 75 + 32a 2
-JW^) + J^(^)
ft'+ftV+ftH
l
< [64]
48« (ft - ft )J
s a a
2 2
In this formula terms of higher & order than ^ and ^ have been 2 2 a a 2 2
neglected. Expanding (64) in terms of — 2 ^- and letting p1
approach p 2 we find for the case of a tube with infinitely thin walls, or of a tube carrying a current of infinitely high frequency,
z
=H( I+ £) log 7- 2 ] ™
66
See Rosa, this Bulletin, 3, p. i; 1907.
67
Grover, Phys. Rev., 30, p. 787; 1910.
Grfve^ Formulas for Mutual and Self-Inductance 113 68 a result which was also found by direct integration, and which was subsequently communicated to us by Mr. T. J. Bromwich. This corresponds to Maxwell's equation (62), but as might be expected gives a slightly greater value for the inductance.
If we expand (64) in terms of ^ and let p approach zero, we find x P2 for the limiting case of a ring with a solid cross section, the same formula (63) as was derived by directly performing the integration for this case. An important case is that of a ring of solid cross section, where the current is not distributed uniformly over the cross section, but the current density is proportional to the distance from the axis of the ring. This would apply to the case of a ring revolving about a diameter in a uniform magnetic field. For this Wien (loc. cit.) derived the formula
Z-4«[(i+|^)logj -093^-1.75 [66]
J. J. THOMSON'S FORMULA FOR RING OF ELLIPTICAL SECTION
If the circular ring has an elliptical section the approximate formula for its self-inductance (corresponding to (59) for a circular 69 section) is
Z= log ^-1.75) [67]
4^J
where a and ft are the semiaxes of the ellipse, and a is the mean radius of the circular ring. The formulas of Minchin, 70 Hicks, 71 and Blathy 72 we have else- where 73 shown to be incorrect, and hence they are not here given.
68 Russell also gives equation (65) but without the term in *L in Phil. Mag., 13,
p. 430; 1907. 69 J. J. Thomson, Phil. Mag., 23, p. 384; 1886. 70 Phil. Mag., 37, p. 300; 1894. 71 Phil. Mag.. 38, p. 456; 1894. 72 Ivondon Electrician, 24, p. 630; Apr. 25, 1890. 73 This Bulletin, 4, p. 149; 1907. H4 Bulletin of the Bureau of Standards [Voi.8,No.i
EXAMPLES ILLUSTRATING THE FORMULAS FOR THE SELF-INDUC- TANCE OF CIRCULAR RINGS OF CIRCULAR SECTION EXAMPLE 52. COMPARISON OF FOUR FORMULAS FOR THE SELF-INDUC- TANCE OF CIRCLES For a circle of radius a = 25 cm and /o = o.c»5 cm we obtain from trie four formulas the following values of L: By Kirchhoff's formula (59) L = 654.4049677 cm By Maxwell's formula (61) L= 654.4053377 cm By Rayleigh and Niven's (63) L= 654.4054877 cm By Wien's second formula (66) L= 654.4061777 cm. Thus for so small a value of — as 1/500 any of these formulas is sufficiently accurate, the greatest difference being less than one in a million, except in the case of formula (66).
EXAMPLE 53. SECOND COMPARISON OF FOUR FORMULAS FOR CIRCLES
For a circle of radius # = 25 cm, /o = o-5 cm, *- being 1/50.
By Kirchhoff's formula (59) Z, = 424. 146477 cm By Maxwell's formula (61) Z = 424. 173477 cm By Rayleigh and Niven's formula (6^) L = 424.178177 cm By Wien's second formula (66) L = 424.232677 cm. EXAMPLE 54. THIRD COMPARISON OF FOUR FORMULAS FOR CIRCLES
For a circle of radius a= 10 cm, r0= 1.0, — = 1/10. a '
By Kirchhoff's formula (59) L= 105.28177 cm By Maxwell's formula (61) L= 105.47677 cm By Rayleigh and Niven's formula (63) L= 105.51777 cm By Wien's second formula (66) Z= 105.90277 cm. It will be seen that for the smallest ring of radius 10 cm and diam- eter of section 2 cm Maxwell's formula gives a result 1 part in 2500 too small, while the simple approximate formula of Kirchhoff is in error by one in four hundred. For the larger rings the differences are much smaller. Wien's second formula gives appreciably larger values than the others, as it should do. GrTver] Formulas for Mutual and Self-Inductance 115
EXAMPLE 55. COMPARISON OF FORMULAS (62) AND (65) FOR VERY THIN WALLED TUBES
(a) a = 25 p = o.o$ cm
By Maxwell's formula (62) Z = 629.4055677 cm
By Formula (65) Z = 629.4057977 cm
Solid ring (63) Z = 654.4054877 cm
(b) a =25 /o = o.5 cm
By Maxwell's formula (62) Z= 399.188977 cm
By Formula (65) Z = 399.206477 cm
Solid ring (63) Z = 424. 178177 cm
(<;) a = 10 /?= 1.0 cm
By Maxwell's formula (62) Z = 95.585^ cm
By Formula (65) Z = 95.71977 cm
Solid ring (63) Z= 105.51777 cm.
Maxwell's expression is nearly correct for the larger ring, but the error increases rapidly as the ratio -^ is increased.
EXAMPLE 56. FORMULA (64) FOR A TUBULAR RING
a = 20 p 2 = 0.5 cm = external radius of the cross section.
The calculation has been carried through for different thicknesses of the walls of the tube (p2 — pt ) ranging from zero (infinitely thin- walled tube) to p 2 (solid cross section).
Pi Pi Z
P* cm
Solid ring 1010.032
0.125 % 1003.210
0.25 % „ 987.528
o-375 V, 968.045
0.5 I Infinitely thin walls 947.308
In formula (64), next to the first two terms, the fourth and fifth terms are the most important. : : :
Ii6 Bulletin of the Bure ate of Standards {Voi.8,No.i
THE SELF-INDUCTANCE OF A SINGLE LAYER COIL OR
SOLENOID
The following approximate formula for the self -inductance of a long solenoid is often given
Z = 4ttVV£ [68]
where a is the mean radius, n is the number of turns of wire per x
cm, and b is the length, supposed great in comparison with a. There is a considerable error in this formula, due to the end effect, but the variations in L due to changes in / are almost exactly pro- portional to the changes in /, and hence this formula may be used for calculating the corresponding variations in L.
RAYLEIGH AND NIVEN'S FORMULAS
The following formula u for the self-inductance of a single layer winding on a solenoid is very accurate when the length b is small compared with the radius a, Fig. 32
L = 4 w{ log
s
- 1 + J^log
£ + 1)) [69]
*f
n is the whole number of turns of wire on the coil, and the radius is measured to the center of the wire. The length b is the mean -- over-all length including the i?zsulatio7t on the first , jo q q <y o o o* anci iasi w {res if the coil is wound closely with insu- lated wire. (See also p. 97.) The self-inductance Ls is, however, not the actual i self-inductance of the coil, but the current sheet value ; that is, it is the value of the self -inductance if the winding were of infinitely thin tape, so that ! the current would cover the entire length b. To get the actual self-inductance L for any given case one must correct L s by formula (80) below. The same remark applies to all the formulas in this section for Ls The approximate formula (68) is too rough to .
make it worth while to apply such a correction.
0000000 For a coil in which the axial dimension b is zero and
Fig. 32 the radial depth is c, the following current sheet for-
mula of Rayleigh and Niven gives the self-inductance 74 Rayleigh's Collected Papers, 2, p. 15. Proc. Roy. Soc, 32, pp. 104-141; 1881. :
Grwer] Formulas for Mutual and Self-Inductance 117
This is not an important case in practice. Formnlas (69) and (70) may be obtained from (88) by making first c=o and then b = o.
COFFIN'S FORMULA
75
Coffin has extended formula (69) so that it is very accurate for coils of length as great as the radius, and sufficiently accurate for most purposes for coils considerably longer than this.
4-H"7:i + ^( '? + 0-i^5(-¥-5) ,
L J
131072 a\
8
b 120/ 4194304 # \ *> b 420/J
LORENZ'S FORMULA
76
L,orenz first gave an exact formula for the self-inductance of a
<, 1) -*
4.0000000000000000000-
1
1
•
1
/
« \yr
1
.
1
1 h/
'O'O'OOOOOOOOOOOOOOOOO
Fig 33
single layer solenoid. It is, like the others, a current sheet formula, and requires correction by (80) for a winding of wire, but applies to a solenoid of any length. Changing the notation slightly Lorenz's formula as originally given is as follows 75 This Bulletin, 2, p. 113; 1906. 76 Wied. Annal., 7, p. 161; 1879. Oeuvres Scientifiques de I,. I^orenz, Tome, 2, i,p. 196. n8 Bulletin of the Bureau of Standards [voi.8, m.i
.2
where £2 = -^| —^ and .F and iT are complete elliptic integrals of the
first and second kind of modulus k, and a ) b, and n are the radius (Fig. 2)3), length, and whole number of turns of wire, respectively. By simple substitutions the formula may be put into the following form, where d is the diagonal of the solenoid = J^c? + b% \
A = ^"W - b*)E+db*F- sA
1
[73]
Coffin derived
77
an expression for L in elliptic integrals which is
equivalent to (73), and also obtained (73) from an expression 78 attributed to Kirchhofl. Formula (73) may be written
L = an
Q
or L = an* Q s
where a is the radius of the solenoid, n is the whole number of
turns on the coil, and Q is the function of — = tan 2,a ( 7) contained in
the square brackets. We have calculated Q for various values of tan 7 from 0.2 to 4.0 and given them in Table IV, page 194. This table will be found useful in calculating L for solenoids when tan 7 s
has one of the values given in the table, as all calculation of elliptic integrals is avoided. In problems where the length and diameter can be chosen at will, as in the designing of apparatus, this method of calculating L will be most frequently useful. The values of the constant Q given in the table have been computed with great care, so that they give very accurate values of Z5 for long as well as ,
short solenoids. In calculating the value of L s by means of formula (69), (71), (73), or (74) and the following, one should use for the length b the over-all
77 This
Bulletin, 2, p. 123, equation (31); 1906.
78 This Bulletin, 2, p. 127, equation (36). The notation is slightly different.
—
Groler] Formulas for Mutual and Self-Inductance 119
length including the insulation (A B, Fig. 34, and not a b) for a close winding of insulated wire, or 71 times the pitch for a uniform winding of bare or covered wire, which is, of course, the same as the length *\ B ,
a c ^ l
from center to center of n + 1 turns. .^^^^^^^^^^^^. The radius a is the mean radius to the ~~^— p r ~ z
center of the wire. The same method ^^^^^^^^^^^^^^B of taking the breadth and depth b and c Fig. 34
applies in the formulas of section 7. (See also remarks under example 47.) NAGAOKA'S FORMULAS AND TABLES In a recent paper 79 Nagaoka has derived formulas and prepared tables by which the self-inductance of a cylindrical current sheet of any dimensions whatever may be accurately and conveniently calcu- lated. Starting from his absolute formula (45) for the mutual inductance of coaxial solenoids, he passes to the special case that the two solenoids coincide, and shows that the resulting expression for the self-inductance is equivalent to Lorenz's absolute formula (72>)i which he then expands in terms of q or q x functions. He expresses the inductance of a coil of finite length by means of the expression (68) for an infinitely long coil, introducing a cor- rection factor K, which is less than unity, to take account of the effect of the ends of the coil. Thus L = \rfa*n?bK= 4ir a jK z % [75]
where K is a function of half the angular aperture of the coil at the center. Nagaoka has prepared tables giving K with as argu-
ment and also as function of the — A -_ = _. These tables are length b reproduced here as Tables XX and XXI, and enable to be obtained K by interpolation with all the accuracy that will usually be required. In case, however, it becomes necessary to obtain a more accurate value of A" than can be obtained from these tables, or in such cases as fall outside the range of the tables, or in a portion where the func- tion changing so rapidly as to make interpolation difficult, the is
following formulas may be used to calculate directly. K 79 Jour. Coll. Sci. Tokyo, 27, art. 6, pp. 18-33; !9°9- 120 Bulletin of the Bttreau of Standards [Vot.s.m.i
For short solenoids
i T k'\ [#'/ 8ft \ 87, |i. 1 "I 4 *
where [76] « = ?.8 + ?i6 + ?i + 1 1!! • • • •
A-ft'+*k*+6ft-+ • • • •
^-^
2
£
7i=^i-4^i + 9^i -
4 9
' * *
•
^ /2 =
4a + £2
2
Sx = 2q -2q + 2q
1 1
i
1
9
+
/1 —
i+V^ (i+^)(i + V^) 8
For relatively long coils
where £ and £' have the same values as in (76) and
/ =
i + V^ (i+^Xi + v* ) 7 1
and a y /3, 7, 8 are given by the same equations as aly /3ly <y 1} Bx in (j6) substituting q in place of q v Table XV will be found convenient
in obtaining q and qx from - and - The more complicated expres- 1 .
2 2
sions for the latter are to be used only when it becomes difficult to obtain 1 — -y/k' and 1 — *yjk without carrying out the calculation of k and k! to an inconvenient number of decimal places. For relatively long coils, for which the angle = tan -1 —r is not greater than 45 °, the simple formula :
Grfver] Formulas for Mutual and Self-Inductance 121
K=\ —^ + 2^+i2? + 44? +ii6/+26o^
2 3 6
+ 57 6^ +
3|o^+. . ..
[ 7 8]
will give values of K correct to a few parts in ten million in the most unfavorable case. The formulas (76), (jj), and (j8) between them cover the entire range of values of 6 with all the precision desired, since the general terms of the series are known. The formula (76) for short coils is the least convenient to use, and for very short coils (69) is preferable. 9 However, by including terms in a in (yy) the range of its applica- bility may be extended to = 8o°, so that (76) need not be used except as a check.
THE WEBSTER-HAVELOCK FORMULA
80
Webster in 1905 by the evaluation of a definite integral, involv-
ing Bessel functions, derived a formula for the inductance of rela- tively long solenoids, which is very simple in form. Havelock 81 gives the same formula as a special application of his formulas for the values of certain integrals of Bessel functions, and stated that 82 the first four terms had already been found by Russell, but seems to have been unacquainted with the work of Webster. This for- mula is
T % a2 n2 \
8 a 1 a2 1 ai 5 a
6
L/yj
64^ 128 £ 10 j
Both Webster and Havelock gave the same expression for the
general term of this series, viz
(-l) s (2s)\(2S+2)\
S\is+2)\{(S+I)\} 2 2s+1 \&
2
80
Bull, of Amer. Math. Soc, 14, No. i, p. 1; 1907.
81
Phil. Mag., 15, p. 332; 1908.
82
Phil. Mag., 13, eq. (48), p. 445; 1907.
122 Bulletin of the Bureau of Standards {Voi.8,No.z
but in all the terms of Webster's final equation (20), except trie first two, a factor 2 has been omitted in the denominator of the coefficients. This expression (79) is in the form adopted by Nagaoka, the expression in the brackets being equivalent to the correction for the ends K tabulated by Nagaoka. ROSA'S CORRECTION FORMULA
Rosa has shown 83 that the above formulas (69 to 79) apply accu- rately only to a winding of infinitely thin strip which completely covers the solenoid (the successive turns being supposed to meet at the edges without making electrical contact) and so realizing the uniform distribution of current over the cylindrical surface which has been assumed in the derivation of all the formulas. A winding of insulated wire or of bare wire in a screw thread may have a greater or less self-inductance than that given by the current sheet formulas above according to the ratio of the diameter of the wire to the pitch of the winding. Putting L for the actual self-inductance of a winding and L8 for the current sheet value given by one of the above formulas, L = L -JL S
The correction JL is given by the following expression: JL = \iran [A + B] [80]
where as above a is the radius, n the whole number of turns of wire and A and B are constants given in Tables VII and VIII, pages 197 and 199. The correction term A depends on the size of the (bare) wire (of diameter d) as compared with the pitch D of the winding; that is, on the value of the ratio d\D. For values of djD less than 0.58, A is negative, and in such cases when the numerical values of A are
greater than the value of B, which is always positive, the correction JL will be negative, and hence L will be greater than Ls (See .
examples 58 and 63.) 63 This Bulletin, 2, pp. 161-187; 1906. :
GrTver] Formulas for Mutual and Self-Inductance 123
THE SUMMATION FORMULA FOR L 84
If we have a single layer winding on a cylinder (Fig. 35), the self-
inductance is equal to the sum of the self-inductances of the separate turns plus the sum of the mutual inductances of each wire on all
the others. Thus, if there are n turns
L=nL + 2(n-i)M + 2(n-2)M + 2(7i-s)M + 1 12 13 li • • + 2Mln [81]
where Lx is the self-inductance of a single turn, 12 is the mutual M inductance of the first and second turns or any two adjacent turns, 1S is the M oooyooo mutual inductance of the first and third or of any two turns separated by one, ^ etc., and m is M the mutual inductance j
of the first and last turns. For a coil {
of four turns this becomes Fig. 35
L = 4Z + 6M + 4M -f 2MU
X 12 13
L should be calculated by formula (63) or any formula for a circular x
ring and M im etc., by (12) or (13). When the number of turns on the coil is small, formula (81) is very convenient, and gives very accurate results. STRASSER'S FORMULA 85 Strasser has derived a formula for the self-inductance of a single layer coil of few turns from (81) by substituting for L its value as x
given by formula (59) and for the various AP$ their values as given by (12). Strasser's formula with slight correction and some changes 86 in notation is as shown on next page
84 Kirchtioff, Gesammelte Abhandlungen, p. 177. 85 Wied. Anna!:, 17, p. 763; 1905.
Strasser uses the formula for L as: L =4wa (log -+0.333 )• This is not quite
•ect. It should be
Z =4iraAog§^-i.75^==47ralog^--i.75+logg8^=4iraAog--|-o.32944y
1
124 Bulletin of the Bureau of Standards [Vol. 8, No. I
L=47ra\ n(log^-i.ys) + n(n-i)(log~-2J-
d% \( 1
8a W(» -i)\2
Jl [82]
where n is trie whole number of turns, d is the pitch, or distance between the centers of two adjacent turns, a is the mean radius of the coil, p is the radius of the section of the wire, and A and B are constants given by Table V, page 195, for values of n up to 30. For coils of a larger number of turns (or indeed any number of turns) the value of L can be accurately calculated by (90) and (93) or by (73) and (80).
SELF-INDUCTANCE OF TOROIDAL COIL OF RECTANGULAR SECTION
The first approximation to the self-inductance of a toroidal coil (that is, a circular solenoid) of rectangular section, wound with a single layer of n turns of wire is
L
s 2n % h log" — [83]
where h is the axial depth of the coil, and rx and r2 are the inner and outer radii of the ring, Fig. 36. Formula (83) is exact for a toroidal
Fig. 36
core enveloped by a current sheet, or for a winding of n turns of infinitely thin tape covering the core completely, the core within G°raveA Formulas for Mutual and Self-Inductance 125
the current sheet being h cm in axial height and (r — r ) cm in 2 t
radial breadth. When the core is wound with round insulated wire, the self- inductance is affected by those lines of force within the cross section of the wire itself, andby those linked with each separate turn of wire in addition to those running through the core. Rosa has shown that the total self-inductance may be more or less than the 87
current sheet value given by (83) according to the size of the wire and the pitch of the winding. In every case, however, the correct value of the self-inductance is derived from the current sheet value L by subtracting a correction term JL, which is equal to twice the s
length of the wire multiplied by the sum of two quantities A and B. Thus L = L -2nl(A+B) s [84]
where n is the whole number of turns in the winding, / is the A is a quantity, depending on the diameter of the length of one turn, wire and the pitch of the winding, given in Table VII, and B is 0.332. When A is negative and greater than B, L is greater than L s. This occurs when the pitch of the winding is more than 2.5 times the diameter of the (uncovered) wire. Frohlich's formula 88 based on the assumption that a winding of round wires is equivalent to a thick current sheet has been shown 89 to be incorrect. CHOICE OF FORMULAS
For a coil of only a few turns the summation formula (81), or Strasser's formula (82) give the inductance with great accuracy without the necessity of correction by Tables VII or VIII. Strasser's formula is, however, accurate only for short solenoids, so that the pitch of the winding can not be very great. For very short solenoids Rayleigh and Niven's formula (69) will give values correct to one in ten thousand for coils whose axial length is as great as one-quarter the diameter of the coil; Coffin's
87
This Bulletin, 4, p. 141; 1907.
88
Wied. Annal., 63, p. 142; 1897.
89
This Bulletin, 4, p. 141; 1907.
21674 — 12 9
126 Bulletin of the Bureau of Standards \yoi.s,No.i
extension of this expression (71) gives as great an accuracy for coils as long as one-half the diameter. These two formulas are probably the most convenient for very short solenoids. For solenoids longer than about one-fifth their diameter the induct- ance may perhaps most readily be calculated by Nagaoka's formula (75), and the Tables XX and XXI. Havelock's formula (79) is accurate and convenient for coils whose axial length is greater than about one and a quarter times the diameter. For purposes of great precision, formulas (76), (yy)^ and (y8) may be used, (76) being indicated for coils shorter than about one-fifth the diameter, (yy) for coils longer than this, and (78) for coils longer than the diameter. L,orenz's absolute formula (73) is of course applicable to coils of all lengths. The interpolation of the elliptic integrals is, however, most easily carried out for coils whose length ranges between one-fifth of the diameter and equality with the latter. The form of this formula is such as to make it neces- sary in some cases to calculate the separate terms to a greater num- ber of places than are required in the result. It must be remembered that all these formulas, with the excep- tion of Strasser's and the summation formula (81) give values for a current sheet, and must be corrected to reduce to the actual wind- ing of round wires. This requires the use of formula (80) and Tables VII and VIII.
EXAMPLES ILLUSTRATING THE FORMULAS FOR THE INDUCTANCE OF SINGLE LAYER SOLENOIDS EXAMPLE 57. RAYLEIGH AND NIVEN'S FORMULA (69) AND CORRECTION FORMULA (80) COMPARED WITH THE SUMMATION FORMULA (81) a = 25 cm, £= 1 cm, n= 10 turns Fig. ^y. Suppose the bare wire is 0.8 mm diameter, the covered wire 1.0 mm. By formula (69) L = 477 X 25 X ioo|log 200 s 6 1
(
log
fe e
200 + - )
2 20,000\ 4/
= 10,000 ttX 4- 79 8 595
= 47,9 8 5-95^ cm
which is the value of L for a current sheet.
G°rover~
Formulas for Mutual and Self-Inductance 127
The correction JL by formula (80) is JL= 1000 ir (A+B) Since D=i.o mm and d=o.S mm, d\D = 0.8 By Table VII, ^=0.3337 " " VIII, ^ = 0.2664 A + B=- 0.6001 ^X = 600.1 7r cm. The value of z/Z, calculated to one place more of decimals is JL = 600. 6 cm 1 7r
L= 47985.95 — 600.l6 77 77
or, £ = 47385.7977- cm. The value of Z may also be calculated by the summation formula (81), using Rayleigh and Niven's formula (63) for L and Maxwell's x
formula (12), for the M y s. The following are the values of the ten terms of (81) and the resulting value of L:
10 A= 6767.19677 cm
l8 M = IO081.664
19
7T
16^3= 7852.535^" 1cm
i ^u= 6303.439^
4
I2M = 5O57.868 7T
-
T
jmm®@mm H
r
15
IO M = 399I.888
16
7T
8M = 3047.78777
17
6M = 2193.46577
1S
4# = 1408.98277
19
AXIS OF CYLINDER.
2 M = 680.982
110
77
Fig. 37
Sum = L = 47385.806 77 cm.
The difference of less than one in a million between the results obtained by formulas (69) and (80) combined and formula (81) is a good check on the corrections of (80), which amount in this
case tomore than 1 per cent of the value of the self-inductance. Formula (69) for as short a coil as this is very accurate, the next term, the fourth term of (71), being inappreciable. If we attempt to use Iyorenz's formula in the above example we notice, first, that 7 is nearly 89 °. The elliptic integrals must con- sequently be calculated by the series formulas (3), which give their value with all the accuracy desired. We meet, however, with the difficulty that the first and third terms are very nearly equal to one J
128 Bulletin of the Bureau of Standards [Voi.8,No,i
another and are several hundred times as large as the second term and the sum of the three terms. Consequently, using seven-place logarithms, it is impossible to obtain the self-inductance closer than about five parts in one hundred thousand. This is also an unfavorable case for (76). Using seven-place logarithms we find
K= 21.281755 — 21.220657 = 0.061098
and consequently L = 47986. 2 77T 8
which is about one part in one hundred thousand larger than the correct value. EXAMPLE 58
As an extreme case to test the use of formulas (69) and (80) we may calculate the self-inductance of a single turn of wire. I,et us take the particular case already calculated by Maxwell's and Ray- leigh and Niven's formulas (61) and (63), example 52. The radius
assume that the wire is covered and that the diameter D is 2 mm. Then — = 0.5. In using Rayleigh's current sheet formula we take
the length of the equivalent current sheet as equal to D. We thus have T L 200 1 0.04 /, 200 i| s * S J 'o.2 2 20000 V 0.2 4/j
l6
\c
y //0 ° — o. s +
= 1007H6.Q077SS —7« I
\
° 500000
I
= 640.777^ cm.
From Tables VII and VIII A = - o. 1363 and B = o. Carrying the value of A to one place of decimals more the value is A = — 0.13628. Thus, since « = i, JL^^ira ( — 0.13628)= — 13.62873-, and being negative is added to Ls . Hence
L= (640.777 + 13.628)^
= 65440571-.
Rosa Grover Formulas for Mutual and Self-Inductance 129
This is identical with the value given by the other formulas, example 52. If we had taken the bare wire of diameter o. 1 cm as equivalent to
a current sheet 0.1 cm long in the above formulas for Ls we should ,
have obtained a different value for Ls but in that case , — would be unity and A would be +.5568. The resulting value of L would, however, be the same as above.
EXAMPLE 59. COFFIN'S FORMULA (71) COMPARED WITH LORENZ'S (73)
We will use for this case a single layer coil wound on an
accurately measured marble cylinder belonging to the Bureau of Standards.
L,ength of winding, /= 30.5510 cm = b in formula (ji>)
Radius " u a = 27.0862 cm
Number of turns n = 440
By (71)
L = 4^440 X 27.0862 1.4590686 + 0.0878241 - 0.0020427
s
+ .0001651 — 0.0000204I
2
= 4^440x27.0862x1.5449947
= 10180999 cm = 0.10180999 henry.
By {73) ^ = 4« + £ = 3868.0128 2 2
4a -£ 2 = 2001.2858
2
7 = 60° 34 43-" 6 55
r
log ^=0.3369388
" £-=0.0811833
Then
Lg== 4^»44Q .150050.12 + 126105.36-158977.00
2
3 (3O.550 I J
or, L = 101810100 cm = 0.10181010 henry. s 1 30 Bulletin of the Bureau of Standards \yoi. 8, No. 1
The correction to be applied to these values is as follows, the diameter of the bare wire being 0.0634 cm, and consequently y,
= 0.9135: .^ = 0.4664
^ = o-3353
(A+B) =0.8017
^1T7ia= 108.3448X44077 = 47671.7 7T
.-. JL= 120067 cm
and £ = 0.10168992 henry by Coffin's formula L = 0.10169003 " by Lorenz's formula. The agreement between these two formulas is very satisfactory, although in Coffin's formula b is greater than a. For shorter coils the accuracy of this formula is better; for longer coils the error rapidly increases.
EXAMPLE 60. NAGAOKA'S FORMULAS (75) AND (77)
We will take for this the coil in the preceding example #= 27.0862 £=30.5510 n = 440 2,a Here —=1.77318 and by interpolation in Table XXI using third differences we find ^=0.557885 — .003165 — .000023 — .000001 = 0.554696 For this case 6 = 6o°34 43. // 655 = 6o?57879, which gives, by /
interpolation in Table XX, K= 0.560382 — .00571 2 4- .000027 ~ -oooooi = 0.554696 Substituting this value of A" in (y^) we find Z,s = o.ioi8ioi3 which differs only three parts in ten million from the value found by Lorenz's formula. Calculating K by (yy) we find
^^=0.70087516
- = 0.087932623
Grller] Formulas for Mutual and Self-Inductance 131
q =0.087943142
^ = 0.007733997
/ = 5-9 8l 5Xio-5
~7
q = 4.626 X 10
6
.\ a = 0.00773446 7 = 0.087703884
/3 = 0.007735385 1 - B = 0.82423335
87 kn
—^-0.37074040
I + = I.OOI4080Q
\-\-a
Sum =1.332 1 4849
multiplied by = I -3°72568 3(1 -BY
^^ = 07525609
.•.^=0.5546959
If we make the calculation with formula (76) 1 + ^=1.93329106 1 --k= 1.87103210 2 log10 £' =1.3825629 lx kn .
"2 '= u.uiyuy/uu
(i+^)(i + v^) 3
== 0.017252703
<7i
<!? = 0.0002976555
=
= 8.86xio~8
^10^ = 1.7631429 .- i log ^= 2.0298933
1+^=1.00029766 1 + ^-1.00238054
871
—^-=0.14295127
1— 01
Sum = 0.46175785
X- log, - = 0.93731902
:
132 Bulletin of the Bureau of Standards [Voi.8,No.i
1 —p = 0-68195058
Sum = 1. 61926960
multiplied by 7=—- = 1.3072565
-t 07525609
2,irk —
.-. ^"=0.5546956
The two formulas give the same value of K withiu about one part in two million. The corresponding- values of L are s
Z = o.ioi8ioio by (yy)
s
Z = o.ioi8ioo5 " (76)
5
the former value being identical with that found by I^orenz's for- mula. This example illustrates well the advantage of obtaining K from Tables XX and XXI rather than by calculation. The accuracy of these tables is ordinarily more than sufficient. The correction to be applied to these current sheet values Ls to obtain the self-inductance Z, is the same as that calculated in the preceding example. EXAMPLE 61. WEBSTER-HAVELOCK FORMULA (79) COMPARED WITH NAGAOKA'S FORMULA (78). LONG COIL a =10 £ = 40 N=^oo and suppose the diameter of the bare wire to be 0.05 cm , 1 a* 2 1.031 25000 2 b
1 a'
—0.00097656
4*
5" _ 6
0.00007629
&
16 b
— ^ — = — O.OOOOO8 2A
a
s
64 b
—tL -—.
10
= O.OOOOO I IO
1 28 b
-23 —ro = — 0.00000008
12
512 b
Sum= 1. 03034241
GrTver] Formulas for Mutual and Self-Inductance 133
a
-= r
—0.21220657
yirb
K= 0.81813584
which, gives
^ = 0.012919483 henry
By (78)
k* = k'* = V^ = 0.94574152
\
1
lik
+ k' = 1. 89442714
•
••
—
2-2(1 + VF)»(i + £')
%
— n m 394
^rn ''Ken
59
q =0.013942860
I+ 2q = 1.02788572
12^ = 0.00233284
44^ = 0.000 1 1926
3
1 16<7* = 0.00000438
26o<f = 0.00000014
5j6q = 0.00000000
6
Sum =1.03034234
— k = 0.21220657
A.
-77
,
^=0.81813577
.*. L = 0.01 2919482 henry
s
which differs by only one part in ten million from the value by the Webster-Havelock formula. The value of K found by interpola- tion in Nagaoka's tables is ^=0.818136. If we solve this problem by means of L,orenz's formula we are
met by the difficulty that 7= 26 °, and therefore the integrals i^and E must be taken from Table XII where their values can not be found more accurately than one part in a million. We find d(^a % -tf)E= — 87909.94 db*F= 118752.95 — Sa = — s 8000.00
Sum= 30843.01
.\L8 = 0.01 291949 henry.
:
134 Bulletin of the Bureau of Standards ivoi.s.No.i
To find the correction to the current sheet value we have — = 0.5,
A= -0.1363
B = +Q-335I
A+B = 0.1
4irna(A+B) = 9999 cm
= 0.00001000 henry,
which must be subtracted from the values of Ls to obtain the self- inductance.
EXAMPLE 62. STRASSER'S FORMULA (82) COMPARED WITH (69) AND (80) AND WITH (81) Take the coil of 10 turns used in example 57 # = 25, d= 0.10 /o = o.o4, n=io. From Table V, ^ = 97.9226 .# = 4241.59 Substituting in (82),
Z=iooJ io(log,^5- 1.75) +9o(log,^- 2) -97.9226
+
,
O.O I \ t
5 ooo(
,
(3log ^-
200
I
N
)
QQOO
2
7r- 424I 59)J
-
or, Z=iooJ 473- 8 3 o6 + °-° 2 75 =473 8 5- 8l7r cm -
This very close agreement with the results by the other two meth- ods (see example 57) is a confirmation of the accuracy of the con- stants A and B of Table V. Of course, a close agreement with (81) is to be expected, for (82) is derived directly from (81).
EXAMPLE 63. FORMULAS (83) AND (84) FOR TOROIDAL COILS
Professor Frohlich's standard of self-inductance had the following dimensions r2 = 35.05377 cm = outer mean radius. rx = 24.97478 cm = inner mean radius. k =20.08455 cm = height, center to center of wire. p =0.011147 cm = radius of wire. n = 2738 = whole number of turns. : :
GrweA Formulas for Mutual and Self-Inductance 135
These values substituted iu (83) give L = 0.1020893 henry. s
The correction AL = — 2nl (A+B) to be substituted in (84) to give the true value of L is found as follows ... The mean spacing of the winding is D = 77-^— — = 0.0689 r -f- r -
The diameter of the bare wire d= 2p =0.0223 .-. d\D =0.324 From Table VII, A= -0.572 B = +0.332 9
.-. A+B = -0.240
2nl= 2X2738 X 60.327 = 330300 cm = whole length of wire in winding. - 2nl(A+B) = + 79,300 cm = 0.0000793 henry L = 0.1020893 " s
L =0.1021686 "
Thus, the correction increases the value of the self-inductance. If the insulation were thinner and the wire thicker (with the same pitch) the correction might be of opposite sign. Thus, if p were 0.02 and hence d\D were 0.58, A would be +0.012 and AL would then be 0.0001130 and L = 0.1019763 henry, considerably less than the preceding value.
THE SELF-INDUCTANCE OF A CIRCULAR COIL OF
RECTANGULAR SECTION
MAXWELL'S APPROXIMATE FORMULA
Maxwell first gave 91 an approximate formula for the important case of a circular coil or conductor of rectangular section, Fig. 38, as follows
L = 47ran (log~-2\
2
[85]
90 This Bulletin, 4, p. 141; 1907. This value applies to any toroidal coils, of 24 turns or more. 91 Elect, and Mag., Vol. II, \ 706. :
i36 Bulletin of the Bureau of Standards [ Vol. 8, No. i
where R is the geometrical mean distance of the cross section of the coil or conductor. The current is supposed uniformly distributed over this section. The value of R for any given shape of rectangular section is given by (i 24) . Its value for several particular cases is
) > given in the table on page 168. It is very nearly
proportional to the perimeter of the rectangle and
1
c approximately equal to 0.2235 (a + j3) where a and
/3 are the length and breadth of the rectangle.
\
Formula (85) is derived from (11) by putting
a
R the geometrical mean distance of the area of
)
the section of the coil from itself, in place of r,
Axis the distance between two circles. If we use (12)
instead of (11) for this purpose, we shall have a
Fig. 38 closer approximation to the value of L. Thus,
R 2
ait an
*5?0 + £M' + i£) [86]
We have placed R in place of d in the second order terms, which 2 2
is ofcourse not strictly correct, as we should use an arithmetical mean square distance instead of a geometrical mean square distance. (See p. 171.) Nevertheless, (86) is a much closer approximation than (85). PERRY'S APPROXIMATE FORMULA Professor Perry has given 92 the following empirical expression for the self-inductance of a short circular coil of rectangular section 2 2 Airn a L [87] 0.2317^+0.44^ + 0.39^
in which n is the whole number of turns of wire, a the mean radius, b the axial breadth, c the radial depth. As in all the formulas of this paper, the dimensions are in centimeters and the value of L is in centimeters. This formula gives a good approximation to L as long as b and c are small compared with a.
John Perry, Phil. Mag., 30, p. 223; 1890.
: :
GroveA Formulas for Mutual and Self-Inductance 137
WEINSTEIN'S FORMULA
Maxwell's more accurate expression for the self-inductance of a 93 circular coil of rectangular section was not quite correct. The 94 investigation was repeated by Weinstein, who gave the following formula Lu = -rran* (X + fi)
where
X=log-
& + --— --log(i+^) + -i- log(i+^) V 2 '
C 12
I2.T % 2
+ —x log ( i + -1) + \x - -) tan-V
%
v
12 * 3 *
[88]
221
/"
= 1 ° g ^~~2 l0g (l
2
+^ ))( +3-^) +345^ +
I
3
9^[(
— i.6irx + 3-2^ tan-1 x z 3 - 2 log (1 + x2 ) + -x* log (1 + i) IO ^v 2 JIT
£ and <; are the breadth and depth of the coil and x = -.
Weinstein's formula for the case of a square section, where b = c reduces to the following simpler expression
Lu = \iranM i+—jlog -^ + .03657-^- [89]
1.194914J
This is a very accurate formula as long as cja is a small quantity. The current is supposed distributed uniformly over the section of the coil, and hence for a winding of round insulated wire, correction must be made by formula (93).
STEFAN'S FORMULA
Stefan 95 simplified by collecting Weinstein's expression (88) together terms depending on the ratio of b to c and computing two short tables of constants yt and y % His formula is as follows: .
z=4 r '
H( i+ w) log vfo"'/i+ ^l [9 ° ]
93
Phil. Trans., 1865, and Collected Works.
94
Wied. Annal., 21, p. 329; 1884.
95
Wied. Annal., 22, p. 113; 1884.
138 Bulletin of the Bureau of Standards \yoi. 8, No. i
The values of y and y are given in Table VI, page 196, as functions x 2
of x = b\c or cjb; that is, x is the ratio of the breadth to the depth of the section, or vice versa, being always less than unity. This formula must be corrected by the quantity J % L as shown below. For the method of taking the dimensions b and c of the cross section, see page 116, section 6; also example 47, page 97.
LONG COIL OF RECTANGULAR SECTION; I. E., SOLENOID OF MORE THAN ONE LAYER ROSA'S METHOD
When the coil is so long that the formula of Stefan is no longer
accurate, the self-inductance may be accurately calculated by a method given by Rosa. 96 In Figs. 39, 40, and 41 are shown three coils, having the same length and mean radius. The first ' f * is a single winding of thin tape and the self-inductance, calculated by a current sheet formula, is Ls .
The second is a single layer of
. wire of square section (length b,
depth c, and b\c turns) and its self-
inductance is L m the current being
supposed uniformly distributed
, . -*„_ over the area of the square con- Fig- 39 ductors. The third is a winding of round insulated wire of length 3, depth and any number of <r,
layers, and its self-inductance is L. These different self-inductances are related as* follows:
L -J L = L U
S X
LU + J L = L
2
. \ L = L - J L + z/ Z
s X 2
L is calculated by any current sheet formula as (69), (71), (72), or s
(JS)- The correction J^L for the depth of the coil is given by the
following formula:
4 L = qiran' \A + B
x 8 9
~
[91]
96
This Bulletin, 4, p. 369; 1907.
Grover] Formulas for Mutual and Self-Inductance 139
This formula has the same form as (80), but some of the quantities have a different meaning; a is the mean radius as before, n is bjc, !
the number of square conductors in the length b, Fig. 40, and A s and B8 are given in Tables IX and X.
pooo
'
\
booo
lOQOQ
a
^v
0000
0000
0000
Fig. 40 Fig. 41
The correction J L is calculated in precisely the same way as %
for a short coil, as described below, formula (93). The above formula for J L gives a very accurate value of the correction to be applied X
to L to obtain L u and permits a test to be made for the error of s ,
Stefan's formula when applied to longer coils than the latter is intended for. Such a calculation shows that for a coil as long as its diameter Stefan's formula (and Weinstein's also, of course) is 1 per cent in error, giving too large a value.
COHEN'S APPROXIMATE FORMULA
Cohen has given the following approximate formula 97 for the self-
inductance of a long coil or solenoid of several layers:
This Bulletin, 4, p. 389; 1907.
140 Bulletin of the Bureau of Standards [ Vol. 8, JVo. 1
L = A.irn*m —1= 2 y2
- — °-
^n^-i)^ + (m — 2)a* + i(v^-r)
-\—hn{m — i)^ 2 + (m — i)(m — 2)a 28 +
2 I
— \m(m — i)a 1
2
+ (;;z — 2)(m — 3)a 2
2
•
where <2 is the mean radius of the solenoid, ati a 2y -. • • a TO are the mean radii of the various layers in the order of their magnitudes, m is the number of layers and Ba is the distance between centers for
any two consecutive layers, and n is the number of turns per unit length. For long solenoids, where the length is, say, four times the diam- eter,we can neglect the last term in equation (92). This formula is sufficiently accurate for most purposes; it will give results accurate to within one-half of 1 per cent even for short solenoids, where the length is only twice the diameter.
MAXWELL'S CORRECTION FORMULA 98
GIVING THE VALUE OF 4zL
Maxwell has shown that when a coil of rectangular section (Fig. 41) is wound with round insulated wire and the self-inductance is calculated by a formula in which the current is assumed to be distrib- uted uniformly over the section, as in Wein- !.•/•>• stein's and Stefan's, the calculated value L u three corrections, each of which mmm*, <^<. is subject to
tends to increase the calculated value of the
§j self-inductance. Thus:
«% %% rA L = L, JL %
D
.
and J^L = Apron log, -= + 0.1 3806 + E \
[93]
Fig. 42
Maxwell showed that the first term takes account of the effect of the insulation, d and D being the diameters of the bare and covered wire, respectively, Fig. 42. The second correction term (0.13806) 98 Elect, and Mag., Vol. II, § 693. GroveA Formulas for Mutual and Self-Inductance 141
reduces from a square section to a circular section for the conductor. The third correction term E takes account of the differences in the mutual inductances of the separate turns of wire on one another when the wire has a round section from what the mutual induc- tances would be if the wire were of square section and no space was occupied by insulation. This term was stated by Maxwell to be equal to — 0.0197 1; it was subsequently stated by Stefan to be equal to + 0.0 1 688. Rosa has shown " that its value is variable, depending on the number of turns of wire in the coil and the shape of the cross section of the latter, and has given the values of E for a number of particular cases. From the following table one can interpolate for E for any par- ticular case not included in the table. Summary of the values of E found for the various cases con- sidered :
2 turns E= . . . 0.006528
3 (one layer) E=
c<
.009045
4
u (two layers) E= .01691
4
u (one layer) E= .01035
8 u (two layers) E= •oi335
10 u (one layer) E= .01276
20 u (one layer) E= .oi357
16 u (four layers) E= .01512
100 (ten layers) E=
(<
.01713
400 a
(20X20) E= .01764
,000
a
(50X20) E= .01778
lfinite number of turns E =
: .01806
The correction J L % is much smaller than z^Z, and can be neglected except when the highest accuracy is sought. The value "This Bulletin, 3, p. 37; 1907. 21674 — 12 10 142 Bulletin of the Bureau of Standards [vol. «?, m. 1
of L and J L can be calculated with accuracy if the dimensions s X
are accurately known, and this is possible if one uses enameled wire of uniform section and takes proper care in winding and measuring the coil. However, such a coil can not be recommended for a standard of the highest precision, and the full theory is given for the sake of completeness and to show the magnitude of the smaller corrections, rather than because all the corrections are likely to be generally needed in practice.
CHOICE OF FORMULAS
If the dimensions of the cross section be very small relatively to
the mean radius, formula (86) may be used. Formula (85) is a still rougher approximation, as is also (87). For somewhat larger cross section Weinstein's formula (88) will give good results. Stefan's form (90) of Weinstein's expression is more convenient to use. Formula (89) is convenient and accurate for coils of square cross section. All these formulas assume that the current is uniformly distributed over the cross section of the coil,and must consequently be corrected by formula (93) to reduce to a winding of round wires. The formulas (88) and (90) begin to be in error for long coils. Cohen's formula (92), however, is most accurate for long solenoids, whose length is more than about four times the diameter. The most accurate formulas are those of Rosa's method (91) and (93). Since the current sheet value may be very accurately obtained by any of the suitable formulas in section 6, this method may be applied to any solenoidal coil whatever. EXAMPLES ILLUSTRATING THE FORMULAS FOR THE SELF-INDUCTANCE OF CIRCULAR COILS OF RECTANGULAR SECTION
EXAMPLE 64. MAXWELL'S APPROXIMATE FORMULAS (85), (86) AND PERRY'S APPROXIMATE FORMULA (87) COMPARED WITH WEIN- STEIN'S FORMULA (89)
Suppose a coil of mean radius 4 cm, with 100 turns of insulated
wire, wound in a square channel 1 X 1 cm. (Fig. 43.) Substituting in (85) a = 4, w=io, ^ = 0.44705 (the g. m. d. of a square 1 cm on a side) we have :
Gaoler] Formulas for Mutual and Self-Inductance 143
Z = 4-4.W[l°g*-^-2]
= 1.141 millihenrys.
This is a first approximation to the self-inductance of the coil. Formula (86) gives a second approximation as follows
= 1 1 46 millihenrys.
.
5=a1
aoooooooocj
Perry's approximate formula, which applies only to C*1 relatively short coils, happens to give a very close ap- proximation for this case. Substituting in (87), the above values, and also b = c= 1, 2
T 47r ioo X 16
a=4»
O.9268+O.44+O.39
= 1.144 millihenrys.
Substituting in the more accurate formula (89) of Weinstein we shall obtain a value with which to com- j
pare the above approximations. Fig. 43
L = i6oooottH 1 + -M log, 22 +0.03657 x ^ - I- i949 I
4J
= 1.147 niillihenrys.
For a = 4, £=2, £=1 /z = 2oo Formula (85) gives 3.750 millihenrys " " " (86) 3.787 (87) " 3.661 (89) « 3.805
Fora=io, &=i, c=i, n=ioo Formula (85) gives 4.005 millihenrys " (86) " 4.007 « (87) " 3.994 " 4.008 " (89)
It will be seen that formula (87) does not give as close approxima- tions as the others, except in the case of the first example, where it .
144 Bulletin of the Bureau of Standards ivoi, 8, No. r
happens to give a value very close to that given by (89). All the values, those of (89) included, are subject to correction by (93) when the coil is wound with round insulated wire.
EXAMPLE 65. FORMULAS (89) AND (90) COMPARED WITH CURRENT- SHEET FORMULAS As a test of these formulas we may calculate the self-inductance of a single turn of wire, using the case already calculated in example
52; that is, a circle of radius a= 25 cm, and the diameter of
p^p 1 the bare wire is 1 mm. Substituting these values in (89) we have
L= iootH +— 3
^ -1.194914
'
( 1 se6 2000 +
)log / 8 y^y
|_\ 15000/ (250) \|
= 640.5995 7T cm.
Substituting in (90),
_TY 01 \l 2o° .OlX.8l62~l
t
L=ioott\ (
LV
n 15000/
,
6 -==- 0.848
)log
-
\/.o2
o
* 340e
—
10000
o
H
>
J
= 640.5995^ cm,
agreeing with the value by (89)
Fig. 44 These values are for a conductor of square cross section (Fig. 44). To reduce to a circular section of same diameter (o. 1 cm) we must apply the second correction term of (93); that is, add to the above value JL = Apra X o. 1 38060 Thus, L= (640.5995 + 13.8060) it = 654.4055^ cm, which agrees with the value found for the self-inductance of a round wire 0.1 cm diameter, bent into a circle of 25 cm radius, by formula (63) example 52 and formulas (69) and (80), example 58.
EXAMPLE 66. STEFAN'S FORMULA (90) COMPARED WITH (69) BY MEANS OF ROSA'S CORRECTION FORMULA (91)
Suppose a coil of mean radius 10 cm, wound with 100 turns in a square channel 1 X 1 cm. Assuming the current uniformly distribu- ted we obtain from (90), in which yx =0.848340,72 =0.8162, :
G°rovtn
Formulas for Mutual and Self-Inductance 145
8a 80
lo
^7^?= ^7j =4 03545lo -
Z m = 4ttX 100,000 fi+-^ J 4.03545-0.84834 + 0.00051 = 4^X3! 8,93° cm = 4.00779 millihenrys. By formula (69) we have for the self-inductance of a current sheet for which a= 10, 6=1, n= 1, A = 4^X38.83475 cm. This is larger than the value for the coil of section iXiby ^Z,, the value of the latter being given by formula (91). By Table IX, ^ = 0.6942. More closely, it is 0.6941 5. 100 By Table X, Bs = o. In this case n1 = 1. Hence, J L = 4?r X 10 X 0.6941 5 = 47r X 6.941 5 cm X
•'•
A=4 7r (38.83475-6.9415) =400.782 cm.
This is the value of the self-inductance for one turn only, the current being uniformly distributed. For 100 turns L is io4 times as great. .*. L u = 4.00782 millihenrys. This value agrees with the above value by Stefan's formula within less than one part in one hundred thousand. For a coil of insulated round wires, this result must be corrected by formula (93). For a coil of the same radius, but of length £= 10 cm, c= 1 cm, wound with 10 layers of 100 turns each, we have the following values By Stefan's formula, y = 0.59243, y = o. 1325 x %
2
Lu = 4ttX 10 x iooo X I-5553 6
= 195.452 millihenrys.
100
This Bulletin, 4, p. 369; 1907.
146 Bulletin of the Bureau of Standards Woi. 8, No. 1
By (69) the current sheet value of L for 10 turns is Z,10 = 4ttX 10 X 100 X 1.65095 = 4^X1650.95. The correction for depth of section by (91) is, since by Tables IX and X, ^ = 0.6942, i? = 0.2792, and therefore A s + Bs = 0.9734 s
4 L = 477-10 X 10 X 0.9734
X
= 47tx 97-34
.-. Lu A - ^Z = 4tt(i 650.95 - 97.34)
=
= 47rX 1553.61 cm for 10 turns.
2
For n= 1000 turns the self-inductance will be ioo times as great.
Z = 4ttXi5-53 6i Xio cm
m
6
= 195.232 millihenrys.
This value is about 1 part in 900 smaller than the above value, showing that Stefan's formula gives too large results by that amount for a coil of this length. If the coil were twice as long, the error
would be about ten times as great. It is interesting to obtain by this method an estimate of the error
by Stefan's formula for coils longer than those for which it is intended. For short coils it is seen to be very accurate, subject always to the corrections of formula (93), and for longer coils it gives a good approximation. The method of (91), however, applies to coils of any length.
EXAMPLE 67. STEFAN'S FORMULA (90) COMPARED WITH (81) AND WITH STRASSER'S (82) FOR COILS OF FEW TURNS, USING THE CORRECTION FORMULA (93) Coil of 2 turns of wire, 0.4 mm diameter, wound in a circle of 1.46 cm radius with a pitch of 2 mm. Stefan's formula assumes a uniform distribution over a rectangular section. Suppose a section as shown in Fig. 45, 4x2 mm, with one turn of wire in the cen- ter of each square. For the rectangular section, with the current uniformly distributed, the self-inductance by Stefan's formula is with a = 1.46, c\b = 0.5, yt = 0.7960, y% = 0.3066, L u = 4.iran z X 2.4763 = 47rtf7zX 4.9526, n being 2. To reduce this to the case of a : —
G?over
Formulas for Mutual and Self-Inductance 147
winding of 2 turns of wire as shown we must apply the corrections given by (93) thus
log D/d= log, 5 = 1.60944
second term =0.13806 j)=s4
.
third term E =0.00653 I • 1 •~|e«2
i-754o
.*. ^2^ = 4^^^X1.7540
L = Lu + 4 L = 47ra7iX 6.7066
2
= 246.1 cm.
By the summation formula (81) we have in this case
L=zL + 2M x 12
-— I t
= \ira [9. 2400 + 4.1 606]
= 245.86 cm.
The value by Strasser's formula is the same as by the summation formula to which it is equivalent. We have also used formulas (69) and (80) for this case and have obtained 246.0. This is one of several problems calculated by Drude 101 by Stefan's formula. Drude concluded that Stefan's formula was inapplicable to such coils, as it gave results from 10 to 25 per cent too large. His trouble was, however, due to taking the length of the coil as the distance between the center of the first wire and the center of the last (instead of n times the pitch) and neglecting the correction terms of formula (93). As we have seen above, Stefan's formula when properly used can be depended upon to give accurate results for short coils, and results within less than 1 per cent for coils of length equal to the radius of the coil. We have calculated several other cases given by Drude and give below the results, together with his experimental values. The radius is the same in each case, and the numbers in the first column are the number of turns in the several coils.
101 wied. Annal., 9, p. 601; 1902.
148 Biclletin of the Bureau of Standards [Vol. 8, No. 1
Drude's Observed
By Stefan's Formula By Rayleigh's By Strasser's Values
n (90) and (93) Formula (69) and (80) Formula (82) or (81) (Values of L in
Centimeters)
2 246.1 246.0 245.9 238.5
4 711.9 711.1 710.8 697.9
6 1298.7 1297.7 1297.8 1271.4
9 2318.0 2313.0 2315.7 2300.1
It will be seen that the values by the different formulas agree very closely, and that the experimental values agree as closely as could be expected for such small inductances.
EXAMPLE 68. FORMULAS (69) AND (80) COMPARED WITH (90) AND (93) FOR COIL OF 20 TURNS WOUND WITH A SINGLE LAYER
a=25 2 cm c = o.i cm 20.
Diameter of bare wire 0.6 mm, of covered wire 1.0 mm. In the last case we obtained the self-inductance of the coil by two distinct methods, the first being the method of summation, the second by assuming the current uniformly distributed over the section, and then applying the three corrections C, E E. In this problem we )
may first calculate L by use of the current sheet formula (69), and then apply the corrections for section, A and B formula (80); and, second, by Stefan's formula for uniform distribution, and apply the three corrections C, E E which give the value for a winding of ) )
round insulated wires. Rayleigh's formula for this example gives:
Z-4WJlog, ioo-o.5 + 2 ^(log ioo+ e
£)J
log, 100 = 4.605170
—-— log, 100 + -) = 0000971
(
20,000 4/ 4.606141
— o. 500000
4.106141
\nran = 40,0007r .\Z, 164 245.64^ cm.
This is the self-inductance of a winding of 20 turns of infinitely thin tape, each turn being 1 mm wide, with edges touching without %°roveA Formulas for Mutual and Self-Inductance 149
making electrical contact, which arrangement fulfills the conditions of a current sheet. To reduce this to the case of round wires we must apply the corrections A and B for self and mutual induction. 102
By Table VII, for d/D = o.6, A = 0.0460
By Table VIII, for n = 20, B = 0.2974
^+^ = 0.3434
^.iran = 2,ooo7r
JL = 47ran(A +B)= 686. 8tt cm
L = L — z/Z=i63 558.84^ cm.
S
By Stefan's formula we find, sustituting the above values of a, n, b, c, and taking ^1 = 0.548990 and y = 0.12 69 2
Lu =i62 234.6o7rcm.
The correction E for a single layer coil of 20 turns is given on page 141. The three corrections are then as follows: C= 0.1 3806 F= o. 5 1082 = log -j- e
^= 0.01357
Sum = 0.66245
.-. JL = 47ran(C+F+JE)= 1324.90^ cm.
.\L = LU + JL= 163 559.507rcm.
This value of L is greater than the value found by the other method by only four parts in a million. Thus we see that the method of calculating Lu by Stefan's or Weinstein's formula and applying the corrections C, i7 , E gives practically identical results with the method of summation and also with the current sheet method for short coils. When, however, the coils are longer, the agreement is not so good, for the reason that the formula of Wein- stein (and Stefan's, derived from it) is not as accurate when the section of the coil is greater. Thus if the coil in the above problem had been 5 cm long and 2.5 mm deep and wound with 20 turns of heavier wire, the difference would have been one part in twenty-five thousand (still very good agreement), and if it were 10 cm long and
102
Rosa, this Bulletin, 2, p. 161; 1906.
1 50 Bulletin of the Bureau of Standards E vol. 8, No. 1
0.5 cm deep (the radius being 25 cm) it would have been one part in two thousand two hundred. For most experimental work, there- fore, Stefan's formula is amply accurate.
EXAMPLE 69. COHEN'S FORMULA (92) COMPARED WITH (91)
A solenoid of length /= 50 cm, mean radius 5 cm, depth of wind- ing 0.4 cm, is wound with 4 layers of wire of 500 turns each. Sub- stituting these values in (92) we have (n = 10) L = i6ttV(i 144.3 + 3336.0 - 10.84 - 1.04) s
= 70.562 millihenrys.
By the second method we first find Ls by (69), then d Lhy (91), x
and J,,L by (93) L = 72.648 millihenrys s
-^Z =-2.167 "
"
AJL = 0.048
L =70.529
This shows a very close agreement between (92) and (91). In calculating Ls we may use Table IV. Since dj I = 0.2 2 2=3.6324, an z = 5x2000 =20,000,000 L = 3.6324 X 20,000,000 cm s
or,
L = 72.648 millihenrys.
s
8. SELF AND MUTUAL INDUCTANCE OF LINEAR
CONDUCTORS 103
SELF-INDUCTANCE OF A STRAIGHT CYLINDRICAL WIRE
The self-inductance of a length / of straight cylindrical wire of radius p is
/
z= 2[/iog ±^±^-V?T?+^+p] [94]
= 2/ log approximately. [95]
Where the permeability of the wire is ^t, and that of the medium outside is unity, (95) appears in the form
Z=2/[log^-I+£] [96]
103
See paper by E. B. Rosa, this Bulletin, 4, p. 301; 1907.
—
Rosa Formulas for Mutual and Self-Inductance
Grover
"I
I5 1
This formula was originally given by Neumann. For a straight cylindrical tube of infinitesimal thickness, or for alternating currents of great frequency, when there is no magnetic field within the wire, the self-inductance is
L = 2l
hH
This is obtained by subtracting from (95) IJ2 or from (96) fi IJ2, [97]
the magnetic flux within the conductor due to unit current.
THE MUTUAL INDUCTANCE OF TWO PARALLEL WIRES
The mutual inductance of two parallel wires of length /, radius />,
and distance apart d is the number of lines of force, due to unit current in one, which cut the other when the current disappears.
This is
M= 2 [98]
. M= 2/ log — — +- approximately
1 [99]
when the length / is great in comparison with d. Equation (98), which is an exact expression when the wires have no appreciable cross section, is not an exact expression for the mutual inductance of two parallel cylindrical wires, but is not appreciably in error even when the section is large and d is small if / is great compared with d.
THE SELF-INDUCTANCE OF A RETURN CIRCUIT
If we have a return circuit of two parallel wires each of length / (the current then flowing in opposite direc- tion in the two wires) the self-inductance of the circuit, neglecting the effect of the end connections shown by dotted lines, Fig. 46, will be very approximately
^Ki-7]
d . /1
M y— ii
Fig. 46
jj
152 Bulletin of the Bureau of Standards Woi. 8, no. r
In the usual case of /*= 1 this will be, when d\l is small
z= 4 /riog ^+ii [101]
If the end effect is large, as when the wires are relatively far apart, use the expression for the self-inductance of a rectangle below (107); or, better, add to the value of (100) the self-inductance of AB + CD, using equation (94) in which /=2AB. Experimental work at the Bureau of Standards, not yet published, has shown that formula (100), and therefore (94) and (98) are con- sistent with the formula (63) for the inductance of a circular ring. [This is equivalent to the following formula in which the loga- rithms are common:
L = 0.741 1 log10 - + .0805 in millihenrys per mile of conductor,
= 0.4605 log - + .050 in millihenrys per kilometer of conductor,
10
d and p being expressed in centimeters, inches, or any other unit.] MUTUAL INDUCTANCE OF TWO LINEAR CONDUCTORS IN THE SAME STRAIGHT LINE The mutual inductance of two adjacent linear conductors of lengths / and m in the same straight line is M = lm I log —r^ + m log m— approximately. I , fio2l
This approximation is very close indeed if the radius of the con- ductor (which has been assumed zero) is very small.
THE SELF-INDUCTANCE OF A STRAIGHT RECTANGULAR BAR
The self-inductance of a straight bar of rectangular section is, to within the accuracy of the approximate formula (99), the same as the mutual inductance of two parallel straight filaments of the same length separated by a distance equal to the geometrical mean dis- tance of the cross section of the bar. Thus,
^/[log^-i+f] [103]
:
Rosa Grover] "I Formulas for Mutual and Self-Inductance 1 53
where J? is trie geometrical mean distance of trie cross section of the rod or bar. If the section is a square, ^ = 0.447 or, a being the side of the square. If the section is a rectangle, the value of R is given by Maxwell's formula (124). This is equivalent to the following
+ - 0.2235O + /3)
2/ 1
L=2l\ log [104]
{ a + /3 ]
In the above formula L is the self-inductance of a straight bar or wire of length / and having a rectangular section of length a and breadth 0. TWO PARALLEL BARS. SELF AND MUTUAL INDUCTANCE The mutual inductance of two parallel straight, square, or rectan- gular bars is equal to the mutual inductance of two parallel wires or filaments of the same length and at a distance apart equal to the geometrical mean distance of the two areas from one another. This is very nearly equal in the case of square sections to the distance between their centers for all distances, the g. m. d. being a very little
x a a
« — «..__>
aVW
Fig. 47
greater for parallel squares, and a very little less for diagonal squares 104 (Fig. 47). We should, therefore, use equation (99) with d equal to g. in. d. of the sections from one another; that is, sub- stantially, to the distances between the centers.
104
Rosa, this Bulletin, 3, p. 1; 1907.
154 Bulletin of the Bureau of Standards \voi. 8 } no.
The self-inductance of a return circuit of two such parallel bars
is equal to twice the self-inductance of one minus twice their mutual inductance. That is, L=2\L -Af\ 1
in which L is calculated by (104) and Mhy (99). x
SELF-INDUCTANCE OF A SQUARE
The self-inductance of a square may be derived from the expres-
sions for the self and mutual inductance of finite straight wires from the consideration that the self-inductance of the square is the sum of the self-inductances of the four sides minus the mutual induc- tances. That is, L = \L -^ x t M the mutual inductance of two mutually perpendicular sides being zero. Substituting a for / and d in formulas (94) and (98) we have,
neglecting p*ja\ L = 8a(log - + £ - .524) [105] p a where a is the length of one side of the square and p is the radius of the wire. If we put /= 4a = whole length of wire in the square,
L=2l( log --f y— 1. 9IO )
or, L=2l( log — 1. 9 10
J
approximately. [ IQ 6]
Formulas (105) and (106) were first given by KirchhofT 105 in 1864.
SELF-INDUCTANCE OF A RECTANGLE
(a) The conductor having a circular section
The self-inductance of the rectangle of length a and breadth b is
L=2{La + L -Ma -M
b h)
where L a and L two sides of length b are the self-inductances of the
a and b taken alone, a and M M h are the mutual inductances of the
two opposite pairs of length a and 6, respectively. From (94) and (98) we therefore have, neglecting p 2 /a2 and put- ,
ting d for the diagonal of the rectangle = -y/a + b* 2
105
Gesammelte Abhandlungen, p. 176. Pogg. Annal., 121, 1864.
Rosa Formulas for Mutual and Self-Inductance Grove) 155
L = 4 (a + b) log — a log (a + a?) — 6 log (0 + a )7
-?(a + 6)+2(^+rtJ [107]
(0) The conductor having a rectangular section
For a rectangle made up of a conductor of rectangular section
Z=4 (a + 6) log^-^-a log (a + a )-^ log (b + d)
7
-^±b 2d+ O.447
+ (* + 0)1 [I08]
where as before a7 is the diagonal of the square. This is equivalent 106 to Sumec's exact formula (6a). For a = 0, a square,
Z = 8ariog^^ + o.2235^? + o.726j [109]
If a = # that is, the section of the conductor is a square,
Z= 8a hog ^ + 0.447^ + 0.033! [no]
MUTUAL INDUCTANCE OF TWO EQUAL PARALLEL RECTANGLES
For two equal parallel rectangles of sides a and b and distance apart d the mutual inductance, which is the sum of the several mutual inductances of parallel sides, is,
M=
*L
al0g
{a + ^+ + b* d*- d )
+ bl °z{ b +
^ +b + d >-~~d^)]
>
+ s\ V<* + 6 + dP- V?+^ - V^+^"+ «*l[iii]
2 2
106
Elektrotech. Zs., 27, p. 1175; 1906.
156 Bulletin of the Bureau of Standards [Vol. 8, no. 1
For a square, where a = 6, we have
+ s[^2d + d -2^a + d' + d'] .
2 2 2 z
[112]
Formula (in) was first given by F. E. Neumann 107 in 1845. The case of two rectangles symmetrically placed about a common vertical axis; the horizontal sides of the smaller rectangle being equi- distant from those of the larger rectangle, has been discussed by Martens 108 and a formula derived which enables the mutual in- ductance to be found for any angle £ between the planes of the rec- tangles. This formula is, however, very elaborate and calculations therewith laborious.
SELF AND MUTUAL INDUCTANCE OF THIN TAPES
The self-inductance of a straight, thin tape of length / and breadth b (and of negligible thickness), Fig. 48 (1), is equal to the mutual inductance of two parallel lines of distance apart R x equal to the geometrical mean distance of the section, which is 0.223136, or
log J?,- log 6-2.
Thus we have approximately Z = 2/[log|-i]
= */[logf+f| [113]
If the thickness of the tape is not negligible, this formula becomes,
when a is the thickness of the tape,
K'-M] c.4]
A closer approximation to L is given by (104), in which a is the thickness and is the breadth of the tape. For two such tapes in the same plane, coming together at their edges with- 107 Allgemeine Gesetze der Inducirteti Strome, Abh. Berlin Akad. 108 Ann. der Phys. 29, p. 963; 1909. —
Rosa Grover Formulas for Mutual and Self-Inductance 157
out making electrical contact, Fig. 48 (2), trie mutual in-
ductance is
M=2l
["5]
= 2 /^log^-o.886 3
J
where R is trie geometrical mean % (1)
distance of one tape from the other, which in this case is 0.892526. For (2)
a return circuit made up of these two tapes the self-inductance is -<- (3) -h -s-
Z=2Z -2^ 1
&<-
= 4^( ^g = 4/ log, 4 [116] j*J (4) (5) = 5.545 X length of one tape. Fig. 48
Thus the self-inductance of such a circuit is independent of the width of the tapes. If the tapes are separated by the distance 6,
Fig. 48 (3), equal to the width of the tapes, ^=1.956536 and L= 8.685/. If the two tapes are not in the same plane, but parallel, Fig. 48 (4),
L = 2L — zM= 4/ log -^
X
[ii7]
and when the distance apart is equal to the breadth of the tapes, Fig. 48 (5) ,we have 7T lo 4;-a and tTT L = A/- ^ = 27T / [118] 2
In this case, also, the self-inductance [217 cm per unit of length] of the pair of thin strips is independent of their width so long as the distance apart is equal to their width. Formula (117) with (132) 21674 — 12- 158 Bulletin of the Bureau of Standards Woi. 8, No. 1
may be employed to calculate the self-inductance of a noninductive shunt made up of a sheet of thin metal doubled on itself.
CONCENTRIC CONDUCTORS
The self-inductance of a thin, straight tube of length / and radius
#s when a % \l is very small, is given by (97), ,
Ar*/[ip*;£-i]
The mutual inductance of such a tube on a conductor within it
is equal to its self-inductance, since all the lines of force due to the outer tube cut through the inner when they collapse on the cessation of current. The self-inductance of the inner conductor, supposed a solid cylinder, is
L = 2l ['-!-!]
x
If the current goes through the latter and returns through the outer tube, the self-inductance of the circuit is
L = L +L - 2M= L - L
X 2 x 2
since M equals Z 2
.: L=2l\~log^ + \~\ [119]
This result can also be obtained by integrating the expression for the force outside ax between the limits a x and a2 and adding the ,
term for the field within a iy there being no magnetic field outside a 2 .
If the outer tube has a thickness a3 — a2 and the current is dis- tributed uniformly over its cross section the self-inductance will be a little greater, the geometrical mean distance from ax to the tube, which is more than a 2 and less than a 8 being given by the ,
expression • _ 2
— a 3loga
-q loga
3
2
2 2
1
1U K a g ~2 -JT~
Putting this value of log a in (119) in place of log a 2 , we should have the self-inductance of the return circuit. If the current is alternating and of very high frequency, the cur- rent would flow on the outer surface of ax and on the inner surface Grfver] Formulas for Mutual and Self-Inductance 159
of the tube, and L for the circuit would be Z=2/los:— ["120I L J at
MULTIPLE CONDUCTORS
If a current be divided equally between two wires of length /, radius p and distance d apart, the self-inductance of the divided conductor is the sum of their separate self-inductances plus twice their mutual inductance. Thus, when d\l is small,
Z = 2/ l0^^"8] =2/ l0g (^i~ [I2I]
[ [
where rgi the g. m. d. of the section of the wire is 0.7788/0 for a round section. If there are three straight conductors in parallel and distance d
apart, the self-inductance is similarly
2
^^pi" 1
]
The expression (rg d )i is the g. m. d. of the multiple conductor. [I22]
EXAMPLES ILLUSTRATING THE FORMULAS FOR THE SELF AND MUTUAL INDUCTANCE OF LINEAR CONDUCTORS * EXAMPLE 70. FORMULAS (94), (95), (96), AND (97) A straight copper wire 100 cm long and 0.2 cm diameter will have a self-inductance by formula (95) of
(200 ^^\)= 1370.18 cm.
logg
0.1 4/
If it were twice as long
L = 400 ( log e j
= 301 7.62 cm.
The more exact formula (94) gives practically the same result where p is so small compared with /. J
160 Bulletin of the Bureau of Standards Woi. 8, No. i
If the wire were of iron with a permeability of iooo, we should
have in the first case for /= ioo
L=200 (log 2000 — i + 250) = 51320 cm.
e
For sufficiently rapid oscillations so that the current may be con- sidered to be confined to the surface of the wire
L= 200 (log 2000— 1) = 1320.18 cm.
e
If the length of the conductor were 10 meters and the diameter 0.2 cm as before, the self-inductance by (95) would be
L = 2000 ( logg 20000 — —1=1 8307.0 cm
= 18.307 microhenrys.
EXAMPLE 71. FORMULAS (98) AND (99)
Two parallel copper wires of length 100 cm and distance apart 200 cm will have a mutual inductance of
M= 2|~iOO loge IQQ +200
IQQ V5 _
lOQ ^
yo + 200 ]
L J
= 2Oo[l0ge!±V5_V5 +
2]
= 200 (loge I- 6l 803— O.2361)
= 49.02 cm.
If the length of each conductor were 200 cm and the distance apart 100 cm, then
^=400 loge 2 + "^5 — V5 + j =330.24 cm.
L I 2 2j
The approximate formula (99) is only applicable when the length of Suppose the conductors is great compared with their distance apart. two conductors 10 meters long are 10 cm apart, then by (99)
2000 10
M= 2000 loge
n/r
& IO
f~i
1 H
,
IOOO
= 2000 [5. 2983 — 0.9900]
= 8616.6 cm = 8.6166 microhenrys.
GrTver] Formulas for Mutual and Self-Inductance 161
The formula (98) gives a value less than two parts in one hundred
thousand greater. EXAMPLE 72. FORMULAS (100) AND (101) Suppose a return circuit of two parallel wires, each 10 meters long and 0.2 cm diameter, distant apart 10 cm, center to center, Fig. 49. The self-inductance of the circuit, neglecting the ends, is by (100) d-10 Hi ZT = 4ooo^log -+--_j IO e IO , I
= 4000X4-8452
= 19380.8 cm= 19.3808 microhenrys.
We have already calculated (example 70) the self-
inductance of one of these two wires by itself. Doubling ^rooo the value we have 36.6140 microhenrys as the self- inductance of two wires in series. In example 71 we calculated the mutual inductance of these two wires. Doubling the value for Mwe have 17.2332 microhenrys. The resultant self-inductance of the circuit (neglecting the ends) is Fig. 49
L=zL — 2^=36.6140— 17.2332
x
= 19.3808 microhenrys.
as found above by formula (100). Taking account of the ends neglected above, we should find that 2,LX for the two ends by (95) is 18 1.9 cm and 2M by (98) is prac- tically zero. Hence the self-inductance of the circuit is, including the ends, L= 19.5627 microhenrys. EXAMPLE 73. FORMULA (102) FOR THE MUTUAL INDUCTANCE OF ADJA- CENT CONDUCTORS IN THE SAME STRAIGHT LINE
When the two conductors are of equal length, l=m, and (102)
becomes M=2llog 2 = 2 /X 0.693 5 cm. e 1
If /= 1000 cm, M= 1386.3 cm. 1 62 Bulletin of the Bureau of Standards Woi. 8, No. i
If m = iooo /, (81) gives
M= I log iooi + iooo / log i.ooi
e
= / log e iooi + / approximately.
If /= i cm, we have M=log iooi + iooo log i.ooi e 6
= 6.909 + 0.999 = 7-9o8 cm.
The self-inductance of the short wire AB, supposed 1 cm long and
of 1 mm radius, is Z= 2 Hog, ^-.75)= 2 (2.9957 -.75) = 4.4915 cm, which is a little more than one-half of the mutual inductance of AB and BC, BC being one thousand times the length of AB. In closed circuits, all the magnetic lines due to a circuit are effective in producing self-inductance, and hence the self-inductance is always greater than the mutual inductance of that circuit with any
other, assuming one turn in each. But with open circuits, as in this case, we may have a mutual inductance between two single con- ductors greater than the self-inductance of one of them.
EXAMPLE 74. FORMULA (104) FOR THE SELF-INDUCTANCE OF A RECTAN- GULAR BAR In formula (164), substituting 1= iooo, and a + /3= 2 for a square bar iooo cm long and 1 square cm section, we have, neglecting the small last term,
Z=2000 [,l0 ^— + 2000 ,
i"|
-
2J
= 2000 (6.908 + 0.5) = 14816 cm
= 14.816 microhenrys.
This would also be the self-inductance for any section having a + /3= 2 cm. EXAMPLE 75. FORMULAS (105) AND (106) FOR THE SELF-INDUCTANCE OF A SQUARE MADE UP OF A ROUND WIRE If the side of the square is 1 meter, a =100 cm, p = o.i cm, we have from (105) Z = 8oo (log 1000 — 0.524) e
= 5107 cm = 5.107 microhenrys.
Grover\ Formulas for Mutual and Self-Inductance 163
If /> = .o5 cm, L = 5662 cm = 5.662 microhenrys. That is, the self-inductance of such a rectangle of round wire is about 1 1 per cent greater for a wire 1 mm in diameter than for one 2 mm in diameter. If Ijp is constant, L is proportional to /, that is, if the thickness of the wire is proportional to the length of the wire in the square, the self-inductance of the square is proportional to its linear dimen- sions.
EXAMPLE 76. FORMULA (107) FOR THE SELF-INDUCTANCE OF A
RECTANGLE OF ROUND WIRE
Suppose a rectangle 2 meters long and 1 meter broad.
Substituting a= 200 cm, b= 100, p = o.i, in (107) we have
£ = 8017.1 cm = 8.01 7 microhenrys.
We can obtain the same result from the values of self and mutual inductances calculated in examples 70 and 71. That is, the result- ant self-inductance of the rectangle is the sum of the self-induct- ances of the four sides, minus twice the mutual inductances of the two pairs of opposite sides. Thus L = A + Z + (L + Z - 2M - 2M ( 3) 2 4) 13 2i
By example 70, L + L = 6035.24
x 3
Z + A= 2740.36 8775.60
2
By example 71, 2Ml3 = 660.48
2M 2,
= 98.04 758.52
.-.Z = 8017.08 cm
= 8.01 71 microhenrys.
The agreement of this result with that obtained from formula
(107) serves as a check on the latter formula, and also illustrates how the values of the self and mutual inductances of open circuits may be combined to give the self-inductance of a closed circuit. —
164 Bulletin of the Bureau of Standards [Vol. 8, no. i
EXAMPLE 77. FORMULAS (108), (109), AND (110) FOR THE SELF-INDUCT- ANCE OF A RECTANGLE OR SQUARE MADE UP OF A BAR OF RECTAN- GULAR SECTION Let a =200 6=100 a = /3=i.ocm.
Substituting these values in (108) we obtain
Z = 4(2971.05- 1209.76-577.95- 150+447.21 +0.99)
= 5926.16 cm.
For a square 10 meters on a side, made of square bar 1 sq cm cross section we have a= 1000, a= 1; substituting in (no)
L = 8000(6.908 + .033)
= 8000 X 6.941 cm = 55.53 microhenrys.
For a circular section, diameter 1 cm, ^ = 0.5; substituting in (105)
L = 8ooo( log,
& 2000 H o. =524 J
\ 2000 /
= 8000 X 7.076 cm = 56.61 microhenrys,
a little more than for a square section, as would be expected.
EXAMPLE 78. FORMULA (112) FOR THE MUTUAL INDUCTANCE OF PARAL- LEL SQUARES
Suppose two parallel squares each 1 meter on a side, 10 cm distant from one another.
a= 100, d=io. Substituting in (112),
J/=8 100 log ( e ,
— J+V20100— 2V i oioo+io
L V1 + V2.01' 1 / J
= 800 log ( e
'-
7==- )+ V 2 01 — 2J1.01 +0.1
-
L \ 1 + V2.01 / J
= 1 142.5 cm= 1. 1425 microhenrys.
EXAMPLE 79. FORMULAS (113), (114), AND (115) FOR THE SELF AND MUTUAL INDUCTANCE OF THIN STRAIGHT STRIPS OR TAPES
Let the tape of thin copper be 10 meters long and 1 cm wide. Grover] Formulas for Mutual and Self-Inductance 165
Substituting 1= 1000 and 6=1 in (113) we have
L = 2000 log 2000 + -
(
e
)
= 2000x8.1009=16202 cm
= 16.202 microhenrys,
as the self-inductance when the conducting strip is very thin. If
the tape is 2 mm thick we may allow for the effect of the thickness by using (114) and we find
L = 2000 X 7.9009 cm = 15.802 microhenrys,
which differs slightly from the preceding value. Two such tapes edge to edge in one plane will have a mutual inductance by (115) of
M= 2000 (log,, 2000 — 0.8863)
= 2000x6.7146 cm
= 13.429 microhenrys.
EXAMPLE 80. FORMULA (117) FOR THE SELF-INDUCTANCE OF A RETURN CIRCUIT OF TWO PARALLEL SHEETS; NONINDUCTIVE SHUNTS
Suppose the dimensions of a thin manganin sheet which has been doubled on itself be as follows:
/= 30 cm £=10 cm d= 1 cm.
By (132) log ^=1.0787
log R = log IO — - = O.8026
x e
Z = 4 /(logi? -logi?
2 1)
= 120X0.2761
= 33.13 cm
= .0331 microhenrys.
EXAMPLE 81. FORMULA (122), 3 CONDUCTORS IN MULTIPLE
Suppose three cylindrical conductors, each 10 meters long and
4 mm diameter, the distance apart of their centers being 1 cm. Substitute in (122) as follows:
/= 1000 cm p= 2 mm <^= 1 cm.
1 66 Bulletin of the Bureau of Standards \yoi. 8, No. i
Then (rg a?)% = 0.538 cm
and L = 2000 log,, ( —1 ) * V 0.538 ) = 2000 X 7.221 cm = 14.442 microhenrys. If the whole current flowed through a single one of the three con- ductors the self-inductance would be
L = 2000 log (
e
- -
J
= 1 7.92 microhenrys,
or about 25 per cent more than when divided among the three.
FORMULAS FOR GEOMETRICAL AND ARITHMETICAL MEAN DISTANCES GEOMETRICAL MEAN DISTANCES Maxwell showed how to calculate mutual and self-inductances in several important cases by means of what he called the geometrical mean distance, either of one conductor from another or of a con- ductor from itself. On account of the importance of this method we give below some of the most useful of these formulas. The geometrical mean distance of a point from a line is the nth root of the product of the n distances from the point P to the various points in the line, n being increased to infinity in deter- mining the value of R. Or, the loga- rithm of R the mean value of log d is
O for
d.
all the infinite values ofthe distance
Similarly, the geometrical mean dis-
tance of a line from itself is the n th root
Fig. 50 of the product ofthe n distances between
all the various pairs ofpoints in the line, n being infinity. ™ 1
Similar definitions apply to the g. m. d. of one area from another,
or of an area from itself.
109
Rosa, this Bulletin, 4, p. 325; 1907.
GrZer] Formulas for Mutual and Self-Inductance 167
The geometrical mean distance R of a line of length a from itself is given by
logR = loga-$
R = ae~ a
* C I2 3]
or R = o.222 >
i2> a
The g. m. d. of a rectangular area of sides a and b from itself is given by
log *=iog v;?TF-i f-I log
^ + g- g j
iog
^|
2& -i«£5
+ £« tan-1
6
+ tan LtJ3
36 »3« 012 [
When the area is a square, and hence a = 6, log* = loga + Ilog 2+ ^-:L5 .-. i? = 0.44705 a ^ For a circular area of radius <z,
log i? = log a — -
4
[126]
,£ = 0.7788 a
For an ellipse of semi-axes a and 6,
logi? = log^-i [127]
A 4
An approximate expression for the g. m. d. of a rectangular area of length a and breadth b is
.£ = 0.2235(0 + 6) [128]
wnich is nearly true for all values of a and b\ that is, the geo- metrical mean distance of the rectangular area from itself is approxi- mately proportional to the perimeter of the rectangle. The following table gives the ratio R\ (a + b) for a series of rectangles of different proportions, from a square to a ratio of 20 to 1 between length and breadth, and finally when the breadth is infinitesimal in comparison with the length. By interpolating for any other case between the — ~ —
i68 Bulletin of the Bureau of Standards [Vol. 8, No. t
values given in the table one can obtain a quite accurate value without the trouble of calculating it by formula (124).
Geometrical Mean Distances of Rectangles of Different Proportions
[a and b are the Length and Breadth, of the Rectangles. R is the Geometrical Mean Distance of its Area]
R
Ratio R a+b
1 :1 0.44705a 0.22353
1.25:1 0.40235a 0.22353
1.5 :1 0.37258a 0.22355
2 :1 0.33540a 0.22360
4 :1 0.27961a 0.22369
10 :1 0.24596a 0.22360
20 :1 0.23463a 0.22346
1 :0 0.22315a 0.22315
The g. m. d. of an annular area of radii a and a 2 from itself is
1
given by 2 _„ a log * = log a - ?{a -a I -#Vv l°g aJ + 4 a x 2 )pJ^f—a 2 x 1 [129]
The g. m. d. of a line of length a from a second line of the same
length, distant in the same straight line na, — ^ — 2 center to center, Fig. 51, is given by the fol- ^ ^ lowing formula: Fig. 51
logJ? n = ± 2 -log(/z + i)a-7i logna + - -log(w- i)a— [!3o] This formula is equivalent to the following, which is more con- 110 venient for calculation for all values of n greater than one.
logJia ^l g„-[-L +^+-±-t+ ^-, + ? ^+ J [131] This formula is very convergent, and only two or three terms are generally required.
Rosa, this Bulletin, 2, p. 168: 1906.
JRosa Formulas for Mutual and Self-Inductance "1
Grover\ 169
The following values of the geometrical mean distances (calling
a unity) were calculated from the above formulas, all after the sec- ond being obtained by (131): ^0 = 0.22313 ^ = 4.98323 ^ = 0.89252 i? 6 = 5.9861O ^ =1.95653 3 i?7 = 6.98806 R = 2.97171 z ^ = 7.98957 R = 3-97 8 90 i ^9 = 8.99076 If the lines are parallel and at distance d, Fig. 52, Fig. 52 the g. m. d. is given by
logJ? = Jlog^ + l(i-J)log(6 2 + ^)+ 2 ^tan-^ [ J 32]
If d= b }
logi? & b+
log --3 [>33]
2 2
The g. m. d. from a point 2, Fig. 53, outside a circle to the circum- ference of the circle, or to the entire area of the circle is the distance dfrom to the center of the circle. 2
(1) The g. m. d. from the center O x to the circumference is of course the radius a. (2) The g. m. d. of any point (as 3) within the circle from the cir- cumference is also a. (3) The g. m. d. of any point on the circumference (as 4 ) from all Fi s- 53 other points of the circumference is also a.
(4) Therefore the g. m. d. of a circular line of radius a from itself is a; that is,
R=a [134]
for each of the four cases named above. The g. m. d. of a point outside a circular ring, Fig. 54, from the ring is the distance d to the center of the ring. The g. m. d. of any point 1} 3, etc., within the ring is given by
logi? =
oi nogo1 -a^ga _ L ? [i35]
The same expression gives the g. m. d. of
any figure, as S x, within the ring from the Fig. 54 ring. The g. m. d. of an external figure, 170 Bulletin of the Bureau of Standards \voi. 8, No. 1
as S 8 , from the annular ring is equal to the g. m. d. of the center X
from the figure S 2 .
The g. m. d. from one circular area to another is the distance between their centers; that is, R=d [136]
for the area Sj with respect to S 8 as it is
Fi 55
for the point O with respect to S
x 2
.
The g. m. d. of a line of length a from a> a second parallel line 01 length a' lo- " ~~f cated symmetrically (Fig. 56) is given by Gray 111 equation (114). , The g. m. d. of 7" a line from a parallel and symmetrically situated rectangle is givenby Gray's equation (112). The g. m. d. of two unequal rectangles from one another is given by Gray's equation (113). 112 The g. m. d. of two adjacent rectangles and of two obliquely situated rectangles are given by Rosa, 113 equations (8a) and (17). As these expressions are somewhat lengthy and not often required they are not repeated here. The values of the g. m. d. for two equal squares in various relative positions to one another have been accu- 114 rately calculated by these formulas, and the results used in the determination 115 of the correction term E of formula (93). 111 Absolute Measurements, Vol. II, Part I. There are a number of misprints in equations 104, 109, ill, and 113 of Gray. The sign of the first term of equation 104 should be +. The signs before^ 2 in the coeffi- cients of the log in the first four terms of equation 109 should be all minus thus % ;
(P -P ),-X(a -p )- A [(<2-i8) -/> ], 2 2 2 2 l 2 2 [(«— a) 2 — /> 2 ]. Similarly in equation +X in the coefficients of the first two terms should be %. (P 2 —P 2 ) ^nd — y2 (a 2 —fi 2 ). In equation 113 the coefficient of /J 4 in each of the first four terms should be /e l
instead of y2 and the first term should have log _{p--b--b / ) 2jrP 2 instead of log ~\
[(p+b+b') 2 -? 2 ].
112
Also by Rosa, equation (8) this Bulletin, 3, p. 6; 1907.
113
This Bulletin, 3, pp. 7 and 12, 1907.
114
This Bulletin, 3, pp. 9-19; 1907.
u5 This Bulletin,
3, p. 37; 1907.
Grovei
Formulas for Mutual and Self-Inductance 171
ARITHMETICAL MEAN DISTANCES
In the determination of self and mntnal inductances by the method 116 of geometrical mean distances it has been shown that more accu- rate formulas can be obtained by the use of certain arithmetical mean distances and arithmetical mean square distances taken in connection with geometrical mean distances. The arithmetical mean distance of a point from a line is the arithmetical mean of the n distances of the point from the various points of the line, n being- infinite. Similarly, the arithmetical mean distance of a line from itself is the arithmetical mean of the distances of the n pairs of points in the line from one another, n being infinite. The a. m. d. of a line of length b from itself is 117
S =-
a [i37]
that is, while the g. m. d. of a line from itself is 0.22313 times its length, the a. m. d. is one-third the length. The arithmetical mean square distance of a Tine from itself is of course larger than the square of the a. m. d. Putting S2 * for the arithmetical mean square distance (a. m. s. d.).
The arithmetical mean distance of a point in the circumference of a circle from the circle is the same as the a. m. d. of the circle from itself; that is, for a circle of radius a,
S = S =*a
1 2 [139]
7T
The arithmetical mean square distance is
S =2a and ^S — a^
2
2 2
2
2
[H ]
(The g. m. d. for this case is R = a, equation (134).) 116 Rosa, this Bulletin, 4, pp. 326-32; 1907. 117 Rosa, this Bulletin, 4, p. 326; 1907. 172 Bulletin of the Bureau of Standards Woi. s, No. 1
The arithmetical mean distance of an external point P from the circumference of a circle, Fig. 57, is
which is the distance PA. The arithmetical mean distance from P to the entire area of the circle is
*=y^+f M Fig. 57
(The g. m. d. for each of these cases is R=d, equation (136).) For the proof of these and other expressions for the arithmetical mean distances and applications of their use see the article referred to above.
10. HIGH-FREQUENCY FORMULAS
Excepting in a very few specified cases, the formulas of the pre- ceding sections apply only to conductors carrying direct current or alternating currents of frequencies so low that the error, due to the assumption that the current is uniformly distributed over the cross section of the wire, is negligible. In the case of standards of mutual inductance the inductance may be regarded as sensibly independent of the frequency, unless the two coils are very close together, and even then the capacity between the coils will be a more potent source of error than the departure of the current from a uniform distribution over the cross section of the wire. The self-inductance of a coil or conductor, on the other hand, depends appreciably on the field in the cross section of the con- ductor, and any deviation of the distribution of the current in the wire from uniformity gives rise to a decrease in the inductance. The amount of this change depends on the frequency of the current and the radius of the cross section of the conductor, as well as on the conductivity and permeability of the material of which it is composed. This decrease of the inductance is accompanied by an increase in the resistance of the conductor. Whereas, however, the inductance Groverl Formulas for Mutual and Self-Inductance 173
with increasing frequency approaches a limiting value, the resistance increases indefinitely as the frequency approaches an infinite value. The change of resistance is always relatively much larger than the change in inductance. The eddy current effects just described are, for the most part, negligible at low frequencies, except in the case of heavy conductors and in coils wound with stout wire in several layers. In the latter case, however, the diminution of the inductance, due to the irregular distribution of the current, is marked, to a greater or less degree, by the effect of the capacity between the windings of the coil, which gives rise to an increase of the inductance with the frequency. For the same reason the resistance is increased more than it would be by the eddy currents alone. Unfortunately, the rigorous or approximate solution of the problem at high frequencies for the various cases for which the inductance with steady currents may be calculated is in many instances very difficult, if Some of the simpler cases, however, not impossible. because of their great importance, have received much attention, with the result that the changes of inductance and resistance may be calculated with a good degree of precision.
STRAIGHT CYLINDRICAL WIRES
This is the most important case of all, since the solution is rigor- ous, and the results may be applied to the construction of practical, absolute standards for high-frequency work. The problem has been treated successively by Maxwell, 118 Heaviside, 119 Rayleigh, 120 and Kelvin. 121 Putting /= length of conductor p = radius of conductor o- = specific resistance of its material fi = permeability f= frequency, p = 2irf R = resistance with current of frequency f f
L' = inductance " " " " /
118
Elect, and Mag., II, § 690.
119
Elect. Papers, II, p. 64.
120
Phil. Mag., 21, p. 381; 1886.
121
Math, and Phys. Papers, III, p. 491; 1889.
21674 — 12 12
174 Bulletin of the Bureau of Standards \yoi. 8, m. i
R = resistance with direct current
L = inductance " " "
llTplX
x
a
Thus R' xW [143] R~ 2 Y
where Z' = 2 {lo g f ^4)] I+ [I44 ]
W= ber x bei' x — bei x ber' x
F=(ber' x)*+(bei' xf
Z= ber x ber' x + bei x bei' x [ x 44 a ]
Since from (96)
we find M log ?- I+ (l
JL = L'-L=-2l l
-
JL
log — i +
J
[146]
For nonmagnetic material the equation (146) takes the form
= " [l47]
-77 \log—
2/
4 -3
In these expressions, ber x and bei x are functions introduced by Lord Kelvin, being respectively the real and imaginary parts of the ordinary Bessel function of order zero, yo having for its argument ,
xi-^ly where x is a real quantity, and i — ^ — 1. These functions are given by the series x
Grover] Formulas for Mutual and Self-Inductance 1 75
2 2 +
ber^r=i --0-0 2 2 2 5
2 4 2 4 6 8 '
[148]
bei^ = -2 --2-2 2 +
2 27 o 2 2 2 2 2
2 2 4 6 '
2 4 6 8 io
and ber';*; and bei .r are their differential coefficients with respect r
to x. These series are very convergent, bnt the calculation, naturally, becomes laborious for large values of x. To lighten the labor of calculation Russell 122 and Savidge 123 have developed asymptotic expressions for ber x, bei x, ber 'x, bei 'x and the auxiliary quanti- ties W, F, and Z which give their numerical values with an accu- y
racy of about one part in ten thousand for values of x greater than about 6, but whose accuracy increases rapidly as x becomes larger. Savidge 123 has, in addition, calculated extensive tables of the above functions and the allied ker and kei functions to four places of decimals, and for values of the argument ranging between 1 and 30 in steps of one unit. These tables will be found very useful in the solution of a variety of problems. For calculation with the formulas (143) to (147), however, it seemed desirable to construct tables in which the argument advances by smaller steps than in the tables of Savidge. For this purpose ber x, bei x, ber x and bei ' '
were calculated directly from their series, for arguments from o. 1 to 5.0, in steps of 0.1, and from 5 to 7 in steps of 0.2. From these were obtained directly by (144a) the values of W, F, Z. For the larger values of x, the quantities W, F, Z were calculated by asymptotic formulas, and checked at a few points by the direct series. Thus the interval from 5 to 10 was covered in steps of 0.2, the interval 10 to 15 in steps of 0.5, and from 15 to 50 in steps of one unit, the aim being to keep the differences of the same order of magnitude. It will, probably, seldom be necessary to make calculations for values of x greater than about 50. If such calculations are occasionally required, they may be made with little trouble by the asymptotic formulas given below.
122
Phil. Mag., 17, p. 524; 1909.
l23
Phil. Mag., 19, p. 49; 1910.
176 Bulletin of the Bureau of Standards [Vol. 8, No. 1
Since making the above calculations, we have determined the expressions for the general terms in Russell's equations (8), (9), and (10), (loc. cit, p. 529), thus materially increasing their range of applicability. These equations, thus extended, are
/fV + jo^ (x\* + i40 (x V +
2
6
ir-±U + , ,
(|3)
3
V^y (|5)
2
V^ (|7)
3
V^
1" -.2
r „2 „2
2~ 3 •
n (\2n-\-
12
(x
Y= x- 3 0' + + -1 +
(|3)"W (&©
v^y
d5)
2
(\7_y\2)
[>49]
1 1-2-3 (2W+l) /*\
+ 2 2 +
(* + i)
2
i 2 3
- • n (|2n
10
l6 l 2 (|3) 3(|5)
;
(f)
+
W [)
2 +
1-2-3 (2^4- 1)
o W
+ (/Z +
+ l) I 2 2 2
2 3
S
71 (\2n+ 1) 2
It would have been simpler to have calculated the values of TV, F, and Z by these formulas than by the more indirect process actually used. The formulas (149) have, however, shown themselves of great service in checking the results. For completeness the asymp- totic formulas of Savidge used have also been appended. They give results to one in one hundred thousand for x> 10 and may be used with an error of not greater than one in ten thousand down to x = 6.
W= 27tx V2
TX\_^2
T (Sx) +
T (8x)\/2 +
%x + JL 75
L ]
x ^'2 6
e Y 9 150
Y= a T (8^) s 4 [150]
d
2irx 8*^3 (8*) ^2 3(8*) ]
= g^r i
3 _ 15 _ _ 45
3 +
315
+
2tt^LV2 8* (S*)\/2 (8^) ^ 2(8^)72
]|
The results are given in Table XXII which gives the values, to one in one hundred thousand, of not only the quantities - -^ and 2 JL GrTver] Formulas for Mutual and Self-Inductance 177
— -^required X I in the preceding formulas, but of -=^r and
These will be found useful in allied problems, and it may seem IX — also.
preferable in some cases to interpolate the values of these latter quantities to obtain the former. For example, with x> 2.5 the first differences, and in some places the second differences also, are smaller
with — than with - -^-. The accuracy of the Table XXII may be regarded as greater than will usually be required, and should suffice for the most precise work. In addition to the general formulas of Kelvin (143) and (144), Rayleigh 124 has given expansions holding for small values of the argument x. These equations, which were extended to another term by Heaviside, are, expressed in the present nomenclature,
=I+
2 F I2\2/ -180V2/ I2.28-30\2/
[151]
4z _ x /*V 12
I 13 (x\* 647 (x\
+ •
xY 24V27 "^4320^2/
2
i2 -36o-56V2/
Their applicability is limited to the range of values of x less than about 2, and it will be more convenient to use Table XXII. For very high frequencies Rayleigh gave also the limiting formulas
In some instances these formulas have been used, as though they were exact, over a considerable range of frequencies, without any statement being made as to the magnitude of the error involved. Expressing these formulas in the present nomenclature, we obtain the following formulas foi infinite frequencies:
m Phil. Mag., 21, p. 387; 1886.
178 Bulletin of the Bureau of Standards \voi. 8, No. x
™~- 4 e ?- + ;(^)]-.
,°
= ^[ lo §-^- I [ J 53]
J
These are seen to be in agreement with equations (143) and (144),
if we remember that the limiting values of — and -=7 TJ/-
as the argu-
ment x is indefinitely increased are both —=• (See formulas (150).)
From (150) we find that only for values of x greater than about 900 is the error from using (152) as small as one-tenth per cent. For x = 70 the error is about 1 per cent, and in many practical cases it is still larger.
The limiting value of the change of inductance is found from
(147) to be
V £/*=« 4 log --3
(JL) x ^=--2 [155]
The error from using (153) is only about one part in ten thousand for x = 60. The error, however, arising from the neglect of the term --^.in (147) is more than 5 per cent.
From (154) we obtain the curious result that the limiting value
of the fractional change of inductance, as the frequency is indefi- nitely increased, depends only on the ratio of the length of the wire to the cross section. Table XXIII gives an idea of the way the limiting value falls off as this ratio is increased. The preceding formulas show that the change of resistance and inductance are functions of the quantity
where K is the conductivity 7 . ;
G°rover] Formulas for Mutual and Self-Inductance 179
Taking the specific resistance of annealed copper at 20 ° as 1.721 microhms or 1721 in absolute electromagnetic units,
A" = 5.8iixio- 4
and (156) takes the simple form
^ = 0.02142/3^/7
To aid in making approximate calculations, and for purposes of orientation, the auxiliary Table XXIV has been calculated, giving the value x = x for copper wire of the above conductivity and of a cross section of 1 mm radius at various frequencies. For the higher frequencies, the corresponding wave length X in meters has been included as likely to be of service in calculations for wireless- telegraph circuits. The range of this table may be considerably
extended by remembering that x varies with -Jfor -. /-. Thus the value of xQ for 7 500 cycles is found directly from the tabulated value for 750 000 cycles by shifting the decimal point. Similarly, the value for A = 150 meters is obtained from the tabulated value for 15 000 meters. It is for this reason that the larger values of X have been tabulated. To calculate x for a copper wire of radius r mm, we have x = x r, and if the conductivity have any value K, the further factor — Finally, if the wire is, in addition, of mag- VI must be applied.
netic material of permeability /*, an additional factor -y//I is necessary to obtain the required value of x.
CONCENTRIC MAIN
The simple case of a cylindrical, straight wire may be regarded as a special case of the more general problem of a concentric main that is, of a solid or tubular inner conductor surrounded by a coaxial tubular outer conductor. This case has been very completely treated by Russell 135 but as the formulas are not simple they are not given ,
here.
125
Phil. Mag., 17, p. 524; 1909.
180 Bulletin of the Bureau of Standards [Vol. 8, No. i
TWO PARALLEL WIRES
Unless the two wires are so near together, relatively to their radins of cross section, that their mutual inductance is appreciably affected by changes in the distribution of the current within the wires, each wire may be treated by the formulas given for a straight, cylindrical wire. Supposing, therefore, that the wires are alike in even* respect
and from (101) we find for wires of nonmagnetic material
JL
L [158]
4 log - + 1
1
d [159]
4 log - + 1
1
P
-I
R'
=
x W [160]
R "2 Y
z
the values of -= and w being taken from Table XXII. — Nicholson 126 has recently given a solution of the problem, when the two wires are so close together that their mutual inductance suffers a sensible change with the frequency. To obtain an idea of the magnitude of this effect, in a practical case, the results by Nichol- son's formulas were compared with those of (158) and (160). With d= 1 cm and p = 0.1 cm, and with a frequency of io 6 , (158) gives
-=— = —8.5 per cent, the effect of Nicholson's correction being to
give a value of -y- numerically only nine parts in ten thousand
smaller.
126
Phil. Mag., IS, p. 417: 1909.
g>Sw] Formulas for Mutual and SelfInductance 1S1
R'
Similarlv for the resistance, \(xS) gives -=- = 7.56, while Xichol-
son's formula reduces this tc 7.55. Since this example relates to a rather unfavorable case, for a standard whose inductance is to be calculated from the dimensions, these corrections for mutual effect may, in general, be regarded as negligible, and the formulas (158 (159), and (160) maybe regarded as sufficiently accurate with the precision usually attainable in the measurement of the dimensions. It is to be noticed that the maximum possible relative change of
inductance, with the frequency, is greater with two parallel wi e s than with either alone, because this change with the parallel wires depends on the sum of their self-inductances which is greater than the resulting self-inductance of the combination (see p. 151). Table
XXIII gives an idea of the values attained by —=- ( J in the case of
the two parallel wires. This maximum change of inductance de- pends only on the ratio of their distance apart to the radius of cross section of the wire. Evidently, other cases of linear conductors of circular cross sec- tion, may likewise be made to depend on the solution for straight wires. CIRCULAR RING OF CIRCULAR SECTION The inductance of a circular ring, in which the current is con- fined wholly to the circumference of the cross section was driven in formula (65). Combining this with (63) we rind that on the imption that (65) represents the actual distribution of the cur- at infinite frequev
The absolute value of the change of inductance ..: infinite fre-
quency is, in the case of a straight wire, see 14
which shows that, if the wire of the ring were stretched ; light, the value given in (161) would become
{JL)^ X - - - ;-.: = — m [16a]
182 Bulletin of the Bureau of Standards \voi.b no.i %
Equation (161) gives, therefore, the effect of the curvatures of the ring, which for ordinary cases will be seen to be small. The resist- ance and inductance of the ring must, therefore, very approximately follow the same law of variation with the frequency as the straight wire. The assumption of formula (65) that at high frequencies the magnetic field is symmetrical around the axis of the cross section of the ring is not strictly true. Actually, it will be a little stronger toward the axis of the ring, so that the amplitude of the current is slightly larger in that part of the cross section which is nearest the axis of the ring. This effect, however, will be extremely small and may be neglected. We have, therefore, with great approximation
2
or, if terms in — may be neglected, 2
L
Hr) [l64]
,
4 log —_7
8a
(*v
\~r)*— \—a - t
l6 5]
4 lo §- 7
r
The values of ( -=— for various values of the determinative \L ),.. )
ratio — are tabulated in Table XXIII. P Neglecting the curvature, the change in resistance will be given by the same expression as for the straight wire, that is R '
^=2^-
x w r Afii
[166]
The quantities
^ - -^ and - ^> are to be taken from Table XXII
2 Y x Y
as before, the argument x being given by (156).
G°rovet
Formulas for Mutual and Self-Inductance 183
EXAMPLES ILLUSTRATING THE FORMULAS FOR HIGH FREQUENCY EXAMPLE 82. STRAIGHT WIRE, VERY HIGH FREQUENCY L,et f= 500000 cycles per second, and ,\ X = 600 meters I = 200 cm /> = 0.125 cm. If the wire is of copper, Table XXIV gives x = 15.146 for a wire of p = o.i cm. We find, therefore, x = 15. 146 X 1.25= 18.932. En- tering Table XXII with this value of x, we find by interpolation, using second differences,
x W R f
4 Z
T = - = 9 5035
,
- 6. ^=0.14923
a slightly more accurate value of the latter may be found by making Z the interpolation for -^.
— =3200 and .*. 4 log — — 3 = 29.284
By (154) ^^=-0.034148 The value found from Table XXIII by interpolation is 0.03416 + dl Formula (147) gives therefore -y^= —0.034148(1 —0.14923) L = —0.029052 By (145) JL = -85.08 cm Recapitulating, the resistance at 500000 cycles per second is 6.95 times as great as with direct current, while the inductance is 85.08 cm or 2.9052 per cent less than the direct current value. This change of the inductance is 85.08 per cent of the possible change of 100 cm (0.034148 of the total inductance). If the wire had been of manganin, for which the conductivity was one thirtieth of that of copper, the value of x becomes
s= 18.932x^^ = 3.4566
184 Bulletin of the Bureau of Standards ivoi. 8, No. 1
and we find — =1.47620 R' , 4Z ±- =.77255
4
I- *y
Z
-.22745
The resistance is 1.47620 times the valne at zero frequency, while the decrease of inductance is only 22.745 P er cen ^ °f the total pos- sible (0.034148), or 0.007767. On the other hand, if the wire had been of iron (conductivity one- seventh of that of copper) and the permeability is assumed as low as 100
x=^iooJ^ 18.932 = 71.556
&
^•=25.551 ^
AZ = 0.039526* 1
-^Z =
4 a
.96047
By (146) (-r)^- "I^p -°-7795o B Y (i45) (^)x=<» = - 10000 cm and the actual changes are 4L = — 9605 cm
\~r)= -0.77950X. 96047= -0.7487
The influence of this relatively low permeability is enormous. The resistance is more than twenty-five times its direct current value, while the inductance is less than the direct current value by nearly 75 per cent of the latter, the maximum possible change with this permeability being about 78 per cent.
EXAMPLE 83. STRAIGHT WIRE—LOW FREQUENCY
If we consider the same wires as in the previous example, except that the frequency is assumed as only 1,000 per second. Then for copper, ^ = 0.6774X1.25 = 0.84675 R' — = 1.00266 rr A.Z nr \Z O.OOI33 ~~y= ^y= O.99867 I Grower] Formulas for Mutual and Self-Inductance 185
JL= —0.133 cm
-j-= -0.034148 (.00133) = - 0.000045
The resistance increase is only 0.266 per cent and the decrease of inductance is only about forty-five millionths of the total.
Formanganin, ^ = 0.84675-* /— JZ-o.: = 0.1582 v 30 By (151) -^ = 1.0000030 2 — =1-0.0000015
1 — -t4Z
—= o. 00000 1 s
xY
The increase in resistance is about three millionths and the decrease in inductance about five hundred-millionths of the direct current values. For iron, with /* = 100 as before
= 0.846757100^ = 3.2004
— = 1.38516 ^=0.81391 1 -4_ =0.18609
JL
-=- = —
100 / /
s —(0.18609) = — 0.14506
o/- \
.
L i28.284 v w ^°
That is, the resistance increase is 38.5 per cent, the inductance decrease 14.5 per cent of the direct current values.
EXAMPLE 84. PARALLEL WIRES
Let us take wires of the same diameter and length as in examples 82 and 83 and consider the same frequencies. The values of
in the previous examples. Further, assume that the distance between the centers of the wires is ^=1.5 cm.
Then —=12 4 log — + 1 = 10.9396 P P : 1
1 86 Bulletin of the Bureau of Standards {Vol. 8, No. r
,\ for nonmagnetic material
(lk\ = ___L_
i
0.091412
\ L/a;=oo IO.9396
as may also be found directly with sufficient precision from Table XXIII. For iron wires, /x= 100 100 = -0.9308. \ L A- 1 d . 109.94 p
The results for the cases treated in the previous examples are, therefore, for the parallel wires, as follows
Material Frequency
R^ Ah
R iv
Copper 500000 6.9504 -0.077772
1000 1.00266 — 0.000122
Manganin 500000 1.4762 — 0.020792
u 1000 — 0.00000014
1.0000030
Iron (/jl= 100) 500000 25-55 1 — 0.8940
1000 1.3852 -0.1732
The following table shows the effect of reducing the radius of cross section to p = o.oi cm AL, Material Frequency R Iv
Copper 50OOOO 1.5 146 0.047522 I.02682 — O.OO0636 " IOOO O.06774 I. OOOOO 1 -2.6XIO-* r Manganin 500000 0.27652 0.047522 I.OOOO30 -7.2XIO— " -3XIO- 13 1000 0.01237 I + 1.2 X IO— Iron o*=ioo) 500000 5.7245 O.833O3 2.2974 0.4273 " " 1000 0.25603 I.000022 -9.3X10-6 EXAMPLE 85. CIRCULAR RING Suppose the ring is of copper and that p = o. 1 cm, a = 20 cm, X = 700 m Then from Table XXIV, *= 14.023 and from Table XXII, R' 4 Z = 5-2173. 1 ' —
Grfver] Formulas for Mutual and Self-Inductance 187
log — = log 1600=7.37776
—p = 0.005
By (162), (z/L)^,^^ — 207r= -62.83 cm. The correction term in (161) =1—0.0000880 By (165) or Table XXIII, =- a °4442 (4^X-» and by (164),
-j-= -0.04442x0.79873
= -0.03548
Washington, January 1, 191 1.
NOTE.
After the present paper had gone to press, a third formula for the mutual induc- tance of coaxial circles was published by Nagaoka (Tokyo Math. Phys., Soc. 6, p. 10; 191 1). This formula was given by Nagaoka in the following form:
• ^-w4^(;:g:g::::;: )
numerator being (— i) n~ 1 n 2 q^
2 ~1
The general term of the and that of the denominator (— i)™(27n+i)<7^—— - M'
The quantity q is calculated from the modulus £/, which is complementary to the modulus k\ of formula (2). Using the same nomenclature as in section 1 of this collection we have
9
,+i
'-i + <9 <9 +
1 — Jk kz
V^r,
k1 _ i~ * _ A^a 2
r r
k,_2
rt + r9 (r. + r,) *
r, + r2
ri = ^/(A + ay + d 2
r ^^A-a) + d 2 2
The general term of the above formula has been given for the sake of complete- ness. In general, however, the convergence is so rapid that all but the first terms are negligible. 1 88 Bulletin of the Bureau of Standards [Vol 8, No. z
As an example of the use of this formula, the calculation for the circles of exam- ples 4 and n above is appended:
A = a = 2$ d=\
^ = 72516 = 50.1 59744 ^2 = 4
4V2516
4 + V2516
^ = 0.72323683
/
- = 0.080303278 ^ = 0.080309959
1 - 40 + 90 = 0.9979281 2
3 s
1 - 2)9 + 5? = 0.98065227
Z 6
log £* = T.i 785770 .-. M= 6o6.o6y6 cm
which agrees very closely with the value 606.0674 found in examples 4 and II. On expansion the above formula becomes
m = 4 VA^{4^q (i + 3q - 4q + 9q - n +
7T
f 2 3 4
l 5
)}
which suggests that the quantity q in this expression is equal to the square of the corresponding quantity in formula (8) above. The truth of this proposition may be established by expressing Landen's transformation in terms of q functions. As regards numerical calculation, therefore, this last formula of Nagaoka is entirely equivalent to his earlier formula (8) . APPENDIX
TABLES OF CONSTANTS AND FUNCTIONS USEFUL IN THE CALCULATION OF MUTUAL AND SELF-INDUCTANCE
21674 — 12 13 189 )
190 Bulletin of the Bureau of Standards [vol. 8, No. 1
TABLE I
Maxwell's Table of
™-^OT=[(H^] 4TT-y/At
(For use with Formula (1)
Log
M,__ Log
M,_
Ax /pr-^Aa
A,
60° 0' 1.499 4780 2 7868 65° 0' 1.637 6633 2 7508 6' 1.502 2648 2 7854 6' 1.640 4141 2 7508 12' / 1.505 0502 2 7840 12 1.643 1649 2 7507 18' 1.507 8342 2 7828 18' 1.645 9156 2 7507 24' 1.510 6170 2 7816 24 / 1.648 6663 2 7507 30' 1.513 3986 2 7803 30' 1.651 4170 2 7509 36' 1.516 1789 2 7790 36' 1.654 1679 2 7510 42' 1.518 9579 2 7778 42' 1.656 9189 2 7512 48 / 1.521 7357 2 7765 48 / 1.659 6701 2 7514 54' 1.524 5122 2 7753 54 / 1.662 4215 2 7516 / 61° 0' 1.527 2875 2 7743 66° 1.665 1731 2 7519 6/ 1.530 0618 2 7734 6' 1.667 9250 2 7522 12' 1.532 8352 2 7725 12' 1.670 6772 2 7524 18' 1.535 6077 2 7715 18 / 1.673 4296 2 7528 24' T.538 3792 2 7705 24 / 1.676 1824 2 7532 30 / 1.541 1497 2 7694 30' 1.678 9356 2 7535 36 / 1.543 9191 2 7683 36 / 1.681 6891 2 7539 42' 1.546 6874 2 7672 42' 1.684 4430 2 7543 48 / 1.549 4546 2 7663 48' 1.687 1973 2 7548 54 / T.552 2209 2 7654 54' 1.689 9521 2 7553 62° 0' 1.554 9863 2 7645 67° 0' 1.692 7074 2 7561 6' 1.557 7508 2 7637 6/ 1.695 4635 2 7567 12' 1.560 5145 2 7629 12 / 1.698 2202 2 7573 18' 1.563 2774 2 7622 18' T.700 9775 2 7580 24' 1.566 0396 2 7615 24 7 1.703 7355 2 7587 30' 1.568 8011 2 7607 30 / 1.706 4942 2 7595 36 / 1.571 5618 2 7598 36 / 1.709 2537 2 7603 42 / T.574 3216 2 7589 42' 1.712 0140 2 7610 48 / 1.577 0805 2 7582 48 7 T.714 7750 2 7619 54' 1.579 8387 2 7575 54' 1.717 5369 2 7628 63° 0' 1.582 5962 2 7570 68° 0' 1.720 2997 2 7637 6' T.585 3532 2 7567 6' 1.723 0634 2 7647 12 / 1.588 1099 2 7563 12' T.725 8281 2 7656 18' 1.590 8662 2 7559 18' 1.728 5937 2 7667 24 / 1.593 6221 2 7555 24' 1.731 3604 2 7679 30 / 1.596 3776 2 7549 30' 1.734 1283 2 7689 36' 1.599 1325 2 7543 36' 1.736 8972 2 7701 42' T.601 8868 2 7537 42' 1.739 6673 2 7713 48' T.604 6405 2 7533 48' 1.742 4386 2 7725 54 / T.607 3938 2 7530 54' 1.745 2111 2 7737 64° 0' 1.610 1468 2 7527 69° 0' 1.747 9848 2 7749 6/ 1.612 8995 2 7524 6' 1.750 7597 2 7763 12' 1.615 6519 2 7521 12' 1.753 5360 2 7778 18 / 1.618 4040 2 7519 18' 1.756 3138 2 7791 24 / 1.621 1559 2 7516 24' 1.759 0929 2 7806 30' 1.623 9075 2 7514 30' 1.761 8735 2 7821 36' T.626 6589 2 7513 36' T.764 6556 2 7836 42' 1.629 4102 2 7512 42' 1.767 4392 2 7853 48" 1.632 1614 2 7510 48' 1.770 2245 2 7871 54 / 1.634 9124 2 7509 54' 1.773 0116 2 7888 65° 0' 1.637 6633 2 7508 70° 0' T.775 8004 2 7904 Rosa "1
Formulas for Mutual and Self-Inductance
Grover\ 191
TABLE I— Continued
Log
M,_ A, Log
M.__ A,
/pr-yjAa
70° 0' 1.775 8004 2 7904 75° 0' 1.918 5141 2 9472 6' 1.778 5908 2 7920 6' 1.921 4613 2 9522 12' 1.781 3828 2 7938 12' 1.924 4135 2 9572 18' 1.784 1766 2 7956 18' T.927 3707 2 9623 24' 1.786 9722 2 7975 24' 1.930 3330 2 9676 30' 1.789 7697 2 7995 30' 1.933 3006 2 9729 36' 1.792 5692 2 8017 36' 1.936 2735 2 9783 42 / 1.795 3709 2 8037 42' 1.939 2518 2 9838 48' 1.798 1746 2 8056 48' 1.942 2356 2 9895 54' 1.800 9802 2 8078 54' 1.945 2251 2 9951 71° 0' 1.803 7880 2 8100 76° 0' T.948 2202 3 0007 6' 1.806 5980 2 8124 6' 1.951 2209 3 0066 12' 1.809 4104 2 8148 12' 1.954 2275 3 0127 18' 1.812 2252 2 8172 18' 1.957 2402 3 0188 24' T.815 0424 2 8195 24' 1.960 2590 3 0251 30' 1.817 8619 2 8220 30' T.963 2841 3 0316 36' 1.820 6839 2 8245 36' 1.966 3157 3 0380 42' 1.823 5084 2 8270 42' 1.969 3537 3 0446 48' 1.826 3354 2 8297 48' 1.972 3983 3 0514 54' 1.829 1651 2 8323 54' 1.975 4497 3 0583 72° 0' 1.831 9974 2 8349 77° 0' 1.978 5080 3 0652 6' 1.834 8323 2 8377 6' 1.981 5731 3 0723 12' 1.837 6700 2 8406 12' 1.984 6454 3 0795 18' 1.840 5106 2 8435 18' 1.987 7249 3 0869 24' 1.843 3541 2 8464 24' 1.990 8118 3 0944 30' 1.846 2005 2 8494 30' 1.993 9062 3 1020 36' T.849 0499 2 8525 36' 1.997 0082 3 1099 42' 1.851 9024 2 8556 42' 0.000 1181 3 1178 48' T.854 7580 2 8588 48' 0.003 2359 3 1259 54' 1.857 6168 2 8620 54' 0.006 3618 3 1341 73° 0' T.860 4788 2 8653 78° 0' 0.009 4959 3 1426 6' 1.863 3441 2 8688 6' 0.012 6385 3 1511 12' 1.866 2129 2 8723 12' 0.015 7896 3 1598 18' 1.869 0852 2 8759 18' 0.018 9494 3 1687 24' 1.871 9611 2 8795 24' 0.022 1181 3 1778 30' 1.874 8406 2 8831 30' 0.025 2959 3 1871 36' T.877 7237 2 8869 36' 0.028 4830 3 1964 42' 1.880 6106 2 8907 42' 0.031 6794 3 2061 48' 1.883 5013 2 8946 48' 0.034 8855 3 2159 54' 1.886 3959 2 8986 54' 0.038 1014 3 2258 74° 0' 1.889 2945 2 9025 79° 0' 0.041 3272 3 2360 6' 1.892 1970 2 9066 6' 0.044 5633 3 2465 12' 1.895 1036 2 9108 12' 0.047 8098 3 2570 18' 1.898 0144 2 9151 18' 0.051 0668 3 2679 24' 1.900 9295 2 9194 24' 0.054 3347 3 2789 30' 1.903 8489 2 9239 30' 0.057 6136 3 2901 36' 1.906 7728 2 9284 36' 0.060 9037 3 3016 42' 1.909 7012 2 9329 42' 0.064 2053 3 3132 48' 1.912 6341 2 9376 48' 0.067 5185 3 3252 54' 1.915 5717 2 9424 54' 0.070 8437 3 3375 75° 0' T.918 5141 2 9472 80° 0' 0.074 1812 3 3500 :
192 Bulletin of the Bureau of Standards [Vol.8, No. 1
TABLE I— Continued
Log
M ,__ A, LOg
M.— Ai
$n-yjAa 47T-vAa
80° 0' 0.074 1812 3 3500 85° 0' 0.265 4154 4 6004 6' 0.077 5312 3 3628 6' 0.270 0156 4 6499 12' 0.080 8940 3 3760 12' 0.274 6655 4 7015 18' 0.084 2700 3 3892 18' 0.279 3670 4 7553 24' 0.087 6592 3 4027 24' 0.284 1223 4 8109 30' 0.091 0619 3 4165 30' 0.288 9332 4 8689 36' 0.094 4784 3 4307 36' 0.293 8021 4 9293 42' 0.097 9091 3 4452 42' 0.298 7314 4 9924 48' 0.101 3543 3 4601 48' 0.303 7238 5 0585 54' 0.104 8144 3 4752 54' 0.308 7823 5 1274 81° 0' 0.108 2896 3 4906 86° 0' 0.313 9097 5 1995 6' 0.111 7802 3 5064 6' 0.319 1092 5 2751 12' 0.115 2866 3 5226 12' 0.324 3843 5 3544 18' 0.118 8092 3 5392 18' 0.329 7387 5 4375 24' 0.122 3484 3 5561 24' 0.335 1762 5 5250 30' 0.125 9045 3 5735 30' 0.340 7012 5 6172 36' 0.129 4780 3 5912 36' 0.346 3184 5 7143 42' 0.133 0692 3 6094 42' 0.352 0327 5 8168 48' 0.136 6786 3 6280 48' 0.357 8495 5 9254 54' 0.140 3066 3 6470 54' 0.363 7749 6 0404 82° 0' 0.143 9536 3 6667 87° 0' 0.369 8154 6 1624 6' 0.147 6203 3 6869 6' 0.375 9777 6 2923 12' 0.151 3072 3 7076 12' 0.382 2700 6 4306 18' 0.155 0148 3 7287 18' 0.388 7006 6 5786 24' 0.158 7435 3 7503 24' 0.395 2792 6 7370 30' 0.162 4938 3 7722 30' 0.402 0162 6 9072 36' 0.166 2660 3 7949 36' 0.408 9234 7 0904 42' 0.170 0609 3 8183 42' 0.416 0138 7 2884 48' 0.173 8792 3 8425 48' 0.423 3022 7 5031 54' 0.177 7217 3 8673 54' 0.430 8053 7 7373 83° 0' 0.181 5890 3 8926 88° 0' 0.438 5417 7 9921 6' 0.185 4816 3 9185 6' 0.446 5341 8 2723 12' 0.189 4001 3 9452 12' 0.454 8064 8 5816 18' 0.193 3453 3 9728 18' 0.463 3880 8 9247 24' 0.197 3181 4 0013 24' 0.472 3127 9 3079 30' 0.201 3194 4 0308 30' 0.481 6206 9 7389 36' 0.205 3502 4 0606 36' 0.491 3595 10 2275 42' 0.209 4108 4 0915 42' 0.501 5870 10 7868 48' 0.213 5023 4 1236 48' 0.512 3738 11 4341 54' 0.217 6259 4 1565 54' 0.523 8079 12 1932 84° 0' 0.221 7824 4 1904 89° 0' 0.536 0011 13 0958 6' 0.225 9728 4 2255 6' 0.549 0969 14 1917 12' 0.230 1983 4 2617 12' 0.563 2886 15 5520 18' 0.234 4600 4 2991 18' 0.578 8406 17 2914 24' 0.238 7591 4 3379 24' 0.596 1320 19 6050 30' 0.243 0970 4 3778 30' 0.615 7370 22 8537 36' 0.247 4748 4 4192 36' 0.638 5907 27 7976 42' 0.251 8940 4 4621 42' 0.666 3883 36 3882 48' 0.256 3561 4 5065 48' 0.702 7765 55 9176 54' 0.260 8626 4 5526 54' 0.758 6941 85° 0' 0.265 4154 4 6004
The above table has been recalculated and some of the values corrected in the last place. The values given are sufficiently accu- rate to give M within one part in a million. A
Rosa Grover} 1 Formulas for Mutual and Self-Inductance *93
TABLE II
Giving the Values of Log F and Log E as Functions of tan y. (See p. 20)
tan? LogF F LogE E
0.1 0.1971 996 1.5747 065 0.1950 415 1.5669 007
0.2 0.2003 678 1.5862 361 0.1918 928 1.5555 817
0.3 0.2054 261 1.6048 192 0.1869 144 1.5378 514
0.4 0.2120 849 1.6296 146 0.1804 536 1.5151 429
0.5 0.2200 096 1.6596 236 0.1729 048 1.4890 346
0.6 0.2288 634 1.6938 051 0.1646 557 1.4610 185
0.7 0.2383 385 1.7311 652 0.1560 492 1.4323 502
0.8 0.2481 728 1.7708 135 0.1473 640 1.4039 900
0.9 0.2581 561 1.8119 912 0.1388 116 1.3766 121
1.0 0.2681 272 1.8540 745 0.1305 409 1.3506 441
1.5 0.3147 473 2.0641 787 0.0955 992 1.2462 329
2.0 0.3535 711 2.2572 057 0.0713 258 1.1784 897
2.5 0.3852 192 2.4278 352 0.0547 850 1.1344 491
3.0 0.4112 984 2.5780 917 0.0432 738 1.1047 748
4.0 0.4518 237 2.8302 429 0.0289 324 1.0688 885
5.0 0.4821 752 3.0351 154 0.0207 426 1.0489 205
7.5 0.5341 061 3.4206 300 0.0109 567 1.0255 497
10.0 0.5682 672 3.7005 581 0.0068 338 1.0158 598
12.5 0.5932 708 3.9198 622 0.0047 004 1.0108 819
TABLE III
Values of the Constant K as Functions of xJA and a/
(For use in Formula (57))
x/A = .50 .75 1 1.25 1.50 1.75 2
a/A
0.50 9.39283 12.30385 14.27982 15.62795 16.56549 17.23299 17.71973 0.55 9.52044 12.40135 14.34594 15.67140 16.59411 17.25215 17.73283 0.60 9.66358 12.50816 14.41766 15.71837 16.62503 17.27286 17.74701 0.65 9.82296 12.62412 14.49474 15.76867 16.65813 17.29504 17.76221 0.70 9.99921 12.74897 14.57688 15.82212 16.69330 17.31865 17.77841 0.75 10.19272 12.88232 14.66377 15.87850 16.73039 17.34357 17.79554 194 Bulletin of the Bureau of Standards \voi.& No.i %
TABLE IV
Values of the Constant Q in Formula (74), Ls =n2aQ
For the self-inductance of a single-layer winding on a solenoid; n is the whole number of turns of wire in the winding and a is the mean radius. The corrections by Tables VII and VIII must be made to get L from Ls as usual. (See p. 122.) In the following table 2a is the diameter, b is the length, and therefore 2#/£ = tan 7. (See Fig. $$.)
2^= tan v Q 2^=tan y
0.20 3.63240 1.80 19.57938
0.30 5.23368 2.00 20.74631
0.40 6.71017 2.20 21.82049
0.50 8.07470 2.40 22.81496
0.60 9.33892 2.60 23.74013
0.70 10.51349 2.80 24.60482
0.80 11.60790 3.00 25.41613
0.90 12.63059 3.20 26.18009
1.00 13.58892 3.40 26.90177
1.20 15.33799 3.60 27.58548
1.40 16.89840 3.80 28.23494
•
1.60 18.30354 4.00 28.85335
For an explanation of the above formula, see page 118. i
Rosa ~
Formulas for Mutual and Self-Inductance
Grover} 195
TABLE V
Constants A and B for Strasser's Formula (82)
A=2log e [0-i)!(/z— 2)! 1] . • • •
B=2>[(n—2)2 loge 2+(n—3)3* loge 3+
2
(n-i) 2 loge (n-i)]
n A B n A B
1 16 354.396 35693
2 17 415.739 46775
3 1.38629 8.318 18 482.75 60314
4 4.96981 46.298 19 555.54 76662
5 11.3259 150.82 20 634.22 96198
6 20.9009 376.05 21 718.89 119330
7 34.0594 794.79 22 809.65 146490
8 51.1097 1499.58 23 906.59 178140
9 72.3189 2603.62 24 1009.81 214760
10 97.9226 4241.59 25 1119.38 256880
11 128.131 6570.33 26 1235.38 305030
12 163.136 9769.51 27 1357.91 359790
13 203.110 14042.2 28 1487.02 421750
14 248.215 19615.3 29 1622.80 491560
15 298.597 26740.1 30 1765.32 569860
We have recently recomputed Strasser's constants, finding several errors which are corrected here. 196 Bulletin of the Bureau of Standards [V0I.8.X0.1
TABLE VI
Table of Constants for Stefan's Formula (90)
»/c or cj b yi y2 &/c or c/ b yi y*
0.00 0.50000 0.1250 0.55 0.80815 0.3437 0.05 .54899 .1269 0.60 .81823 .3839 0.10 .59243 .1325 0.65 .82648 .4274 0.15 .63102 .1418 0.70 .83311 .4739 0.20 .66520 .1548 0.75 .83831 .5234 0.25 .69532 .1714 0.80 .84225 .5760 0.30 .72172 .1916 0.85 .84509 .6317
0.35 .74469 .2152 0.90 .84697 .6902
0.40 .76454 .2423 0.95 .84801 .7518
0.45 .78154 .2728 1.00 .84834 .8162
0.50 .79600 .3066
There is in general no difficulty in obtaining yx and y % with suffi- cient accuracy by interpolation from this table, using second and in some cases third differences. The only case which may give trouble is when b\c or cjb is less than 0.1. In such cases, however, Stefan's formula does not give precise results, and the errors in the interpolation will not be important. Rosa Formulas for Mutual and Self-Inductance "1
Grover J 197
TABLE VII
Values of Correction Term A , Depending on the Ratio — of the Diameters of Bare and Covered
Wire on the Single Layer Coil
(For use in Formula (80))
A=log c I .74 5 2-
( D)
d d d D A A, D A A, D A A, A,
j
1.00 0.5568 -100 0.75 0.2691 -134 l 0.50 -0.1363 -202 - 4 .99 .5468 -101 .74 .2557 -136 .49 - .1565 -206 - 5 .98 .5367 -103 .73 .2421 -138 | .48 - .1771 -211 - 4 .97 .5264 -104 .72 .2283 -140 .47 - .1982 -215 - 4 .96 .5160 -105 .71 .2143 -142 .46 - .2197 -219 - 6 0.95 0.5055 -106 0.70 0.2001 -144* 0.45 -0.2416 -225 - 5 .94 .4949 -107 .69 .1857 -146 .44 - .2641 -230 - 5 .93 .4842 -108 .68 .1711 -148 .43 - .2871 -235 - 6 .92 .4734 -109 .67 .1563 -150 .42 - .3106 -241 - 6 .91 .4625 -110 .66 .1413 -152 .41 - .3347 -247 - 6 0.90 0.4515 -112 0.65 0.1261 -155 0.40 -0.3594 -253 - 7 .89 .4403 -113 .64 .1106 -157 .39 - .3847 -260 - 7 .88 .4290 -114 .63 .0949 -160 .38 - .4107 -267 - 7 .87 .4176 -116 .62 .0789 -163 .37 - .4374 -274 - 7 .86 .4060 -117 .61 .0626 -166 .36 - .4648 -281 - 9 0.85 0.3943 -118 0.60 0.0460 -168 0.35 -0.4929 -290 - 9 .84 .3825 -120 .59 .0292 -171 .34 - .5219 -299 - 9 .83 .3705 -121 .58 + .0121 -174 .33 - .5518 -308 - 9 .82 .3584 -123 .57 - .0053 -177 .32 - .5826 -317 - 9 .81 .3461 -124 < .56 - .0230 -180 .31 - .6143 -328 -11
0.80 0.3337 -126 0.55 -0.0410 -184 0.30 -0.6471 -339 -12 .79 .3211 -127 .54 - .0594 -187 .29 - .6810 -351 -13 .78 .3084 -129 .53 - .0781 -190 .28 - .7161 -364 -13 .77 .2955 -131 .52 - .0971 -194 .27 - .7525 -377 -15 .76 .2824 -133 .51 - .1165 -198 .26 - .7902 -392 -16 0.75 0.2691 0.50 -0.1363 0.25 -0.8294 198 Bulletin of the Bureau of Standards I Vol. 8, No. 1
TABLE VII— Continued
d d D A Ai A 3 D A Ai A*
0.25 -0.8294 -408 - 18 0.10 -1.7457 -1054 - 124 .24 - .8702 -426 - 19 .09 -1.8511 -1178 - 157 .23 - .9128 -445 - 20 .08 -1.9689 -1335 - 206 .22 - .9573 -465 - 23 .07 -2.1024 -1541 - 283 .21 -1.0038 -488 - 25 .06 -2.2565 -1824 - 407 0.20 -1.0526 -513 - 28 0.05 -2.4389 -2231 - 646 .19 -1.1039 -541 - 30 .04 -2.6620 -2877 -1177 .18 -1.1580 -571 - 35 .03 -2.9497 -4054 -2878 .17 -1.2151 -606 - 39 .02 -3.3551 -6932 .16 -1.2757 -645 - 45 .01 -4.0483
0.15 -1.3402 -690 - 51 .14 -1.4092 -741 - 60 .13 -1.4833 -801 - 70 .12 -1.5634 -870 - 83 .11 -1.6504 -953 -101 0.10 -1.7457 ) )
Rosa 1 Formulas for Mutual and Self-Inductance Grover\ 199
TABLE VIII
Values of the Correction Term B, Depending on the Number of Turns of Wire on the Single Layer Coil
(For use in Formula (80)
B=-
n
2 mloge
z
—
Km
where Rm is geometric mean distance of the sections of the current sheet whose centers coincide with those of the wires. (See this Bulletin, 2, p. 168, equat. (11); 1 906,
Number ol Turns =n B Number of Turns= n B
1 0.0000 50 0.3186
2 .1137 60 .3216
3 .1663 70 .3239
4 .1973 80 .3257
5 .2180 90 .3270
6 .2329 100 .3280
7 .2443 125 .3298
8 .2532 150 .3311
9 .2604 175 .3321
10 .2664 200 .3328
15 .2857 300 .3343
20 .2974 400 .3351
25 .3042 500 .3356
30 .3083 600 .3359
35 .3119 700 .3361
40 .3148 800 .3363
45 .3169 900 .3364
50 .3186 1000 .3365
200 Bulletin of the Bureau of Standards [ J 'o!.S . No. i
TABLE IX
Value of the Constant A s as a Function of t\a, t being the Depth of the Winding and a the Mean Radius
^= 6^-^(loge 8f+^.76)
-
(For use in Formula (91))
t/a A 3
0.6949
0.10 0.6942
0.15 0.6933
0.20 0.6922
0.25 0.6909
TABLE X
Values of the Correction Term Bs depending on the Number of Turns of Square Conductor on Single Layer Coil
(For use in Formula (91))
n i \ -Km /
where i? /TO =geom. mean distance for the two squares Rm = " ii «« elements of the current sheet. (See this Bulletin, 4, p. 373; 1907.)
Number of B Number of Number of Turns s Turns B 3 Turns B 8
1 0.0000 11 0.2844 21 0.3116
2 .1202 12 .2888 22 .3131
3 .1753 13 .2927 23 .3145
4 .2076 14 .2961 24 .3157
5 .2292 15 .2991 25 .3169
6 .2446 16 .3017 26 .3180
7 .2563 17 .3041 27 .3190
8 .2656 18 .3062 28 .3200
9 .2730 19 .3082 29 .3209
10 .2792 20 .3099 30 .3218
Rosa Formulas for Mutual and Self-Inducta?tce Grover 201
TABLE XI
Table of Napierian Logarithms to Nine Decimal Places for Numbers from 1 to 100
1 0.000 000 000 51 3.931 825 633
2 0.693 147 181 52 3.951 243 719
3 1.098 612 289 53 3.970 291 914
4 1.386 294 361 54 3.988 984 047
5 1.609 437 912 55 4.007 333 185
6 1.791 759 469 56 4.025 351 691
7 1.945 910 149 57 4.043 051 268
8 2.079 441 542 58 4.060 443 Oil
9 2.197 224 577 59 4.077 537 444
10 2.302 585 093 60 4.094 344 562
11 2.397 895 273 61 4.110 873 864
12 2.484 906 650 62 4.127 134 385
13 2.564 949 357 63 4.143 134 726
14 2.639 057 330 64 4.158 883 083
15 2.708 050 201 65 4.174 387 270
16 2.772 588 722 66 4.189 654 742
17 2.833 213 344 67 4.204 692 619
18 2.890 371 758 68 4.219 507 705
19 2.944 438 979 69 4.234 106 505
20 2.995 732 274 70 4.248 495 242
21 3.044 522 438 71 4.262 679 877
22 3.091 042 453 72 4.276 666 119
23 3.135 494 216 73 4.290 459 441
24 3.178 053 830 74 4.304 065 093
25 3.218 875 825 75 4.317 488 114
26 3.258 096 538 76 4.330 733 340
27 3.295 836 866 77 4.343 805 422
28 3.332 204 510 78 4.356 708 827
29 3.367 295 830 79 4.369 447 852
30 3.401 197 382 80 4.382 026 635
31 3.433 987 204 81 4.394 339 155
32 3.465 735 903 82 4.406 719 247
33 3.496 507 561 83 4.418 840 608
34 3.526 360 525 84 4.430 816 799
35 3.555 348 061 85 4.442 651 256
36 3.583 518 938 86 4.454 347 296
37 3.610 917 913 87 4.465 908 119
38 3.637 586 160 88 4.477 336 814
39 3.663 561 646 89 4.488 636 370
40 3.688 879 454 90 4.499 809 670
41 3.713 572 067 91 4.510 859 507
42 3.737 669 618 92 4.521 788 577
43 3.761 200 116 93 4.532 599 493
44 3.784 189 634 94 4.543 294 782
45 3.806 662 490 95 4.553 876 892
46 3.828 641 396 96 4.564 348 191
47 3.850 147 602 ( 97 4.574 710 979
48 3.871 201 Oil 98 4.584 967 479
49 3.891 820 298 99 4.595 119 850
50 3.912 023 005 100 4.605 170 186
log i525=log 25+log 61; log 9. 8=log 98— log 10, etc.
:
202 Bulletin of the Bureau of Standards {Vol. 8, JSTo. 2
TABLE XII
Values of F and E
The following table of elliptic integrals of the first and second kind is taken from Legendre's Traite des Fonctions Elliptiques, Volume 2, Table VIII
F A, A2 E &i A2
0° 1.570 796 120 239 0° 1.570 796 - 120 -239 1 1.570 916 359 240 1 1.570 677 - 359 -239 2 1.571 275 599 240 2 1.570 318 - 598 -239 3 1.571 874 839 241 3 1.569 720 - 836 -238 4 1.572 712 1 080 241 4 1.568 884 -1 075 -238 5 1.573 792 1 321 243 5 1.567 809 -1 312 -237 6 1.575 114 1 564 244 6 1.566 497 -1 549 -236 7 1.576 678 1 808 246 7 1.564 948 -1 785 -235 8 1.578 486 2 054 247 8 1.563 162 -2 020 -234 9 1.580 541 2 302 249 9 1.561 142 -2 255 -233 10 1.582 843 2 551 252 10 1.558 887 -2 487 -232 11 1.585 394 2 803 254 11 1.556 400 -2 719 -230 12 1.588 197 3 057 257 12 1.553 681 -2 949 -228 13 1.591 254 3 314 260 13 1.550 732 -3 177 -227 14 1.594 568 3 574 263 14 1.547 554 -3 404 -225 15 1.598 142 3 836 266 15 1.544 150 -3 629 -223 16 1.601 978 4 103 270 16 1.540 521 -3 852 -221 17 1.606 081 4 373 274 17 1.536 670 -4 073 -218 18 1.610 454 4 647 278 18 1.532 597 -4 291 -216 19 1.615 101 4 925 283 19 1.528 306 -4 507 -214 20 1.620 026 5 208 288 20 1.523 799 -4 721 -211 21 1.625 234 5 495 293 21 1.519 079 -4 932 -208 22 1.630 729 5 788 298 22 1.514 147 -5 140 -205 23 1.636 517 6 087 304 1 23 1.509 007 -5 345 -202 24 1.642 604 6 391 311 i 24 1.503 662 -5 547 -199 25 1.648 995 6 702 317 ! 25 1.498 115 -5 746 -196 26 1.655 697 7 019 324 i 26 1.492 368 -5 942 -T92 27 1.662 716 7 343 332 27 1.486 427 -6 134 -189 28 1.670 059 7 675 340 28 1.480 293 -6 323 -185 29 1.677 735 8 015 349 29 1.473 970 -6 508 -181 30 1.685 750 8 364 358 1 30 1.467 462 -6 689 -177 31 1.694 114 8 722 367 1 31 1.460 774 -6 866 -173 32 1.702 836 9 089 377 32 1.453 908 -7 039 -168 33 1.711 925 9 466 388 33 1.446 869 -7 207 -164 34 1.721 391 9 854 400 34 1.439 662 *-7 371 -159 35 1.731 245 10 254 412 | 35 1.432 291 -7 531 -155 36 1.741 499 10 666 425 1 36 1.424 760 -7 685 -150 37 1.752 165 11 091 439 37 1.417 075 -7 835 -145 38 1.763 256 11 530 453 38 1.409 240 -7 980 -140 39 1.774 786 11 983 469 39 1.401 260 -8 120 -134 40 1.786 770 12 452 486 40 1.393 140 -8 254 -129 41 1.799 222 12 938 504 41 1.384 886 -8 382 -123 42 1.812 160 13 442 523 42 1.376 504 -8 505 -117 43 1.825 602 13 965 543 43 1.367 999 -8 622 -111 44 1.839 567 14 508 565 44 1.359 377 -8 733 -105 45 1.854 075 15 073 588 45 1.350 644 -8 838 - 98 Rosa Groverj 1 Formulas for Mutual and Self-Indtcctance 203
TABLE XII— Continued
F A, A 2
E Ai A a
45° 1.854 075 15 073 588 45° 1.350 644 -8 838 -98 46 1.869 148 15 661 613 46 1.341 806 -8 936 -92 47 1.884 809 16 274 640 47 1.332 870 -9 028 -85 48 1.901 083 16 914 669 48 1.323 842 -9 113 -78 49 1.917 997 17 584 700 49 1.314 729 -9 190 -71 50 1.935 581 18 284 735 50 1.305 539 -9 261 -63 51 1.953 865 19 017 770 51 1.296 278 -9 324 -56 52 1.972 882 19 787 809 52 1.286 954 -9 380 -48 53 1.992 670 20 597 852 53 1.277 574 -9 427 -40 54 2.013 266 21 449 898 54 1.268 147 -9 467 -31 55 2.034 715 22 347 949 55 1.258 680 -9 498 -22 56 2.057 062 23 296 1 004 56 1.249 182 -9 520 -14 57 2.080 358 24 300 1 064 57 1.239 661 -9 534 - 4 58 2.104 658 25 364 1 130 58 1.230 127 -9 538 + 5 59 2.130 021 26 494 1 203 59 1.220 589 -9 533 +15 60 2.156 516 27 698 1 284 60 1.211 056 -9 518 +25 61 2.184 213 28 982 1 373 61 1.201 538 -9 492 36 62 2.213 195 30 355 1 472 62 1.192 046 -9 457 47 63 2.243 549 31 827 1 583 63 1.182 589 -9 410 58 64 2.275 376 33 410 1 708 64 1.173 180 -9 351 70 65 2.308 787 35 118 1 848 65 1.163 828 -9 281 82 66 2.343 905 36 965 2 006 66 1.154 547 -9 199 95 67 2.380 870 38 971 2 186 67 1.145 348 -9 104 109 68 2.419 842 41 158 2 393 68 1.136 244 -8 995 123 69 2.460 999 43 551 2 631 69 1.127 250 -8 872 138 70 2.504 550 46 181 2 907 70 1.118 378 -8 734 153
71 2.550 731 49 088 3 230 71 1.109 643 -8 581 169 72 2.599 820 52 318 3 611 72 1.101 062 -8 412 187 73 2.652 138 55 930 4 066 73 1.092 650 -8 225 205 74 2.708 068 59 996 4 614 74 1.084 425 -8 020 224 75 2.768 063 64 609 5 283 75 1.076 405 -7 796 245 76 2.832 673 69 892 6 112 76 1.068 610 -7 550 268 77 2.902 565 76 004 7 156 77- 1.061 059 -7 282 292 78 2.978 569 83 160 8 497 78 1.053 777 -6 990 318 79 3.061 729 91 657 10 261 79 1.046 786 -6 672 347 80 3.15* 385 101 918 12 647 80 1.040 114 -6 325 379 81 3.255 303 114 565 15 989 81 1.033 789 -5 946 415 82 3.369 868 130 554 20 879 82 1.027 844 -5 531 455 83 3.500 422 151 433 28 453 83 1.022 313 -5 076 502 84 3.651 856 179 886 41 130 84 1.017 237 -4 573 558 85 3.831 742 221 016 64 880 85 1.012 664 -4 016 626 86 4.052 758 285 896 118 167 86 1.008 648 -3 389 715 87 4.338 654 404 063 288 129 87 1.005 259 -2 675 842 88 4.742 717 692 193 88 1.002 584 -1 832 1081 89 5.434 910 89 1.000 752 - 752 90 90 1.000 000 204 Bulletin of the Bureau of Standards [ Vol. 8, No. i
TABLE XIII
Values of log F and log E
(See Note, p. 213)
y LogF Aj A, LogE Aj A a
45?0 0.2681 2722 3 4688 105 0.1305 4086 2 8279 52 45.1 0.2684 7411 3 4793 105 0.1302 5807 2 8331 52 45.2 0.2688 2204 3 4898 105 0.1299 7476 2 8383 52 45.3 0.2691 7102 3 5004 106 0.1296 9094 2 8434 52 45.4 0.2695 2106 3 5110 106 0.1294 0659 2 8486 51
45.5 0.2698 7216 3 5216 106 0.1291 2174 2 8537 51 45.6 0.2702 2431 3 5322 106 0.1288 3636 2 8589 51 45.7 0.2705 7753 3 5428 107 0.1285 5048 2 8640 51 45.8 0.2709 3181 3 5535 107 0.1282 6408 2 8691 51 45.9 0.2712 8716 3 5642 107 0.1279 7717 2 8742 51
46.0 0.2716 4358 3 5749 108 0.1276 8975 2 8793 51 46.1 0.2720 0108 3 5857 108 0.1274 0182 2 8844 51 46.2 0.2723 5965 3 5965 108 0.1271 1338 2 8894 50 46.3 0.2727 1930 3 6073 108 0.1268 2444 2 8945 50 46.4 0.2730 8003 3 6181 109 0.1265 3499 2 8995 50 46.5 0.2734 4184 3 6290 109 0.1262 4504 2 9045 50 46.6 0.2738 0474 3 6399 109 0.1259 5459 2 9095 50 46.7 0.2741 6873 3 6508 110 0.1256 6364 2 9145 50 46.8 0.2745 3381 3 6618 110 0.1253 7218 2 9195 50 46.9 0.2748 9999 3 6728 110 0.1250 8023 2 9245 50 47.0 0.2752 6727 3 6838 110 0.1247 8778 2 9295 49 47.1 0.2756 3565 3 6948 111 0.1244 9483 2 9344 49 47.2 0.2760 0513 3 7059 111 0.1242 0139 2 9393 49 47.3 0.2763 7572 3 7170 111 0.1239 0746 2 9443 49 47.4 0.2767 4741 3 7281 112 0.1236 1303 2 9492 49 47.5 0.2771 2023 3 7393 112 0.1233 1811 2 9541 49 47.6 0.2774 9415 3 7505 112 0.1230 2271 2 9589 49 47.7 0.2778 6920 3 7617 112 0.1227 2681 2 9638 49 47.8 0.2782 4537 3 7729 113 0.1224 3043 2 9687 48 47.9 0.2786 2266 3 7842 113 0.1221 3357 2 9735 48 48.0 0.2790 0109 3 7955 113 0.1218 3622 2 9783 48 48.1 0.2793 8064 3 8069 114 0.1215 3838 2 9831 48 48.2 0.2797 6133 3 8183 114 0.1212 4007 2 9879 48 48.3 0.2801 4315 3 8297 114 0.1209 4128 2 9927 48 48.4 0.2805 2612 3 8411 115 0.1206 4201 2 9975 48
48.5 0.2809 1023 3 8526 115 0.1203 4226 3 0022 47 48.6 0.2812 9548 3 8641 115 0.1200 4204 3 0070 47 48.7 0.2816 8189 3 8756 116 0.1197 4134 3 0117 47 48.8 0.2820 6945 3 8872 116 0.1194 4017 3 0164 47 48.9 0.2824 5817 3 8988 116 0.1191 3854 3 0211 47 49.0 0.2828 4805 3 9104 117 0.1188 3643 3 0258 47 49.1 0.2832 3909 3 9221 117 0.1185 3385 3 0304 46 49.2 0.2836 3130 3 9338 117 0.1182 3081 3 0351 46 49.3 0.2840 2467 3 9455 118 0.1179 2730 3 0397 46 49.4 0.2844 1923 3 9573 118 0.1176 2333 3 0443 46
49.5 0.2848 1495 3 9691 118 0.1173 1890 3 0489 46 49.6 0.2852 1186 3 9809 119 0.1170 1401 3 0535 46 49.7 0.2856 0996 3 9928 119 0.1167 0866 3 0581 46 49.8 0.2860 0924 4 0047 119 0.1164 0286 3 0626 45 49.9 0.2864 0971 4 0167 120 0.1160 9660 3 0671 45 50.0 0.2868 1137 4 0286 120 ! 0.1157 8988 3 0717 ! 45 Grller] Formulas far Mutual and Self-Inductance 205
TABLE XIII— Continued
V LogF Ai A 3 LogE Ai A2
50?0 0.2868 1137 4 0286 120 0.1157 8988 3 0717 45 50.1 0.2872 1424 4 0406 121 0.1154 8271 3 0762 45 50.2 0.2876 1830 4 0527 121 0.1151 7510 3 0807 45 50.3 0.2880 2357 4 0648 121 0.1148 6703 3 0851 45 50.4 0.2884 3005 4 0769 122 0.1145 5852 3 0896 44 50.5 0.2888 3774 4 0891 122 0.1142 4956 3 0940 44 50.6 0.2892 4665 4 1013 122 0.1139 4016 3 0985 44 50.7 0.2896 5677 4 1135 123 0.1136 3032 3 1028 44 50.8 0.2900 6812 4 1258 123 0.1133 2003 3 1072 44 50.9 0.2904 8070 4 1381 123 0.1130 0931 3 1116 43 51.0 0.2908 9451 4 1504 124 0.1126 9815 3 1159 43 51.1 0.2913 0955 4 1628 124 0.1123 8656 3 1203 43 51.2 0.2917 2584 4 1753 125 0.1120 7453 3 1246 43 51.3 0.2921 4336 4 1877 125 0.1117 6207 3 1289 43 51.4 0.2925 6214 4 2002 125 0.1114 4919 3 1332 43 51.5 0.2929 8216 4 2128 126 0.1111 3587 3 1374 42 51.6 0.2934 0344 4 2254 126 0.1108 2213 3 1417 42 51.7 0.2938 2597 4 2380 127 0.1105 0796 3 1459 42 51.8 0.2942 4977 4 2506 127 0.1101 9337 3 1501 42 51.9 0.2946 7483 4 2634 127 0.1098 7836 3 1543 42 52.0 0.2951 0117 4 2761 128 0.1095 6294 3 1584 41 52.1 0.2955 2878 4 2889 128 0.1092 4709 3 1626 41 52.2 0.2959 5767 4 3017 129 0.1089 3083 3 1667 41 52.3 0.2963 8784 4 3146 129 0.1086 1416 • 3 1708 41 52.4 0.2968 1930 4 3275 130 0.1082 9707 3 1749 41
52.5 0.2972 5205 4 3405 130 0.1079 7958 3 1790 41 52.6 0.2976 8610 4 3535 130 0.1076 6168 3 1831 40 52.7 0.2981 2144 4 3665 131 0.1073 4338 3 1871 40 52.8 0.2985 5810 4 3796 131 0.1070 2467 3 1911 40 52.9 0.2989 9606 4 3927 132 0.1067 0556 3 1951 40 53.0 0.2994 3533 4 4059 132 0.1063 8605 3 1991 40 53.1 0.2998 7592 4 4191 133 0.1060 6614 3 2030 39 53.2 0.3003 1783 4 4324 133 0.1057 4584 3 2070 39 53.3 0.3007 6107 4 4457 134 0.1054 2514 3 2109 39 53.4 0.3012 0564 4 4591 134 0.1051 0406 3 2148 39 53.5 0.3016 5155 4 4725 134 0.1047 8258 3 2186 38 53.6 0.3020 9880 4 4859 135 0.1044 6072 3 2225 38 53.7 0.3025 4739 4 4994 135 1041 3847 3 2263 38 53.8 0.3029 9733 4 5130 136 0.1038 1584 3 2301 38 53.9 0.3034 4863 4 5265 136 0.1034 9283 3 2339 38 54.0 0.3039 0128 4 5402 137 0.1031 6944 3 2377 37 54.1 0.3043 5530 4 5539 137 0.1028 4567 3 2414 37 54.2 0.3048 1069 4 5676 138 0.1025 2153 3 2451 37 54.3 0.3052 6745 4 5814 138 0.1021 9702 3 2488 37 54.4 0.3057 2559 4 5952 139 0.1018 7214 3 2525 37 54.5 0.3061 8511 4 6091 139 0.1015 4689 3 2562 36 54.6 0.3066 4602 4 6230 140 0.1012 2127 3 2598 36 54.7 0.3071 0833 4 6370 140 0.1008 9529 3 2634 36 54.8 0.3075 7203 4 6511 141 0.1005 6895 3 2670 36 54.9 0.3080 3714 4 6652 141 0.1002 4226 3 2705 35 55.0 0.3085 0365 4 6793 142 0.0999 1520 3 2741 35
216 74 14
2o6 Bulletin of the Bureau of Standards Woi. <?, No, i
TABLE XIII— Continued.
7 LogF Ai A. LogE Ai A2
55?0 0.3085 0365 4 6793 142 0.0999 1520 3 2741 35 55.1 0.3089 7158 4 6935 142 0.0995 8779 3 2776 35 55.2 0.3094 4093 4 7077 143 0.0992 6003 3 2811 35 55.3 0.3099 1170 4 7220 143 0.0989 3193 3 2846 34 55.4 0.3103 8391 4 7364 144 0.0986 0347 3 2880 34 55.5 0.3108 5754 4 7508 145 0.0982 7467 3 2914 34 55.6 0.3113 3262 4 7652 145 0.0979 4553 3 2948 34 55.7 0.3118 0915 4 7798 146 0.0976 1605 3 2982 33 55.8 0.3122 8712 4 7943 146 0.0972 8623 3 3015 33 55.9 0.3127 6655 4 8089 147 0.0969 5607 .3 3049 33 56.0 0.3132 4745 4 8236 147 0.0966 2559 3 3082 33 56.1 0.3137 2981 4 8384 148 0.0962 9477 3 3114 32 56.2 0.3142 1365 4 8532 149 0.0959 6363 3 3147 32 56.3 0.3146 9896 4 8680 149 0.0956 3216 3 3179 32 56.4 0.3151 8577 4 8829 150 0.0953 0037 3 3211 32 56.5 0.3156 7406 4 8979 150 0.0949 6826 3 3243 31 56.6 0.3161 6385 4 9129 151 0.0946 3583 3 3274 31 56.7 0.3166 5514 4 9280 151 0.0943 0309 3 3305 31 56.8 0.3171 4794 4 9432 152 0.0939 7003 3 3336 31 56.9 0.3176 4226 4 9584 153 0.0936 3667 3 3367 30 57.0 0.3181 3809 4 9736 153 0.0933 0300 3 3397 30 57.1 0.3186 3545 4 9890 154 0.0929 6903 3 3428 30 57.2 0.3191 3435 5 0044 155 0.0926 3475 3 3457 30 57.3 0.3196 3479 5 0198 155 0.0923 0018 3 3487 29 57.4 0.3201 3677 5 0353 156 0.0919 6531 3 3516 29 57.5 0.3206 4030 5 0509 156 0.0916 3014 3 3545 29 57.6 0.3211 4539 5 0666 157 0.0912 9469 3 3574 28 57.7 0.3216 5204 5 0823 158 0.0909 5895 3 3603 28 57.8 0.3221 6027 5 0980 158 0.0906 2292 3 3631 28 57.9 0.3226 7008 5 1139 159 0.0902 8662 3 3659 28 58.0 0.3231 8146 5 1298 160 0.0899 5003 3 3686 27 58.1 0.3236 9444 5 1458 160 0.0896 1317 3 3714 27 58.2 0.3242 0902 5 1618 161 0.0892 7603 3 3741 27 58.3 0.3247 2520 5 1779 162 0.0889 3862 3 3767 26 58.4 0.3252 4299 5 1941 162 0.0886 0095 3 3794 26 58.5 0.3257 6240 5 2104 163 0.0882 6301 3 3820 26 58.6 0.3262 8344 5 2267 164 0.0879 2481 3 3846 26 58.7 0.3268 0611 5 2431 165 0.0875 8635 3 3871 25 58.8 0.3273 3041 5 2595 165 0.0872 4764 3 3897 25 58.9 0.3278 5637 5 2761 166 0.0869 0867 3 3922 25 59.0 0.3283 8397 5 2927 167 0.0865 6945 3 3946 24 59.1 0.3289 1324 5 3094 168 0.0862 2999 3 3971 24 59.2 0.3294 4418 5 3261 168 0.0858 9028 3 3995 24 59.3 0.3299 7679 5 3429 169 0.0855 5033 3 4018 23 59.4 0.3305 1108 5 3598 170 0.0852 1015 3 4042 23 59.5 0.3310 4707 5 3768 171 0.0848 6973 3 4065 23 59.6 0.3315 8475 5 3939 171 0.0845 2908 3 4088 22 59.7 0.3321 2414 5 4110 172 0.0841 8820 3 4110 22 59.8 0.3326 6524 5 4282 173 0.0838 4710 3 4132 22 59.9 0.3332 0806 5 4455 174 0.0835 0578 3 4154 21 60.0 0.3337 5261 5 4629 175 0.0831 6424 3 4176 21 Rosa "I Grover\ Formulas for Mutual and Self-Inductance 207
TABLE XIII— Continued
V LogF A, A2 LogE Ai A 2
60?0 0.3337 5261 5 4629 175 0.0831 6424 3 4176 21 60.1 0.3342 9890 5 4803 175 0.0828 2248 3 4197 21 60.2 0.3348 4694 5 4979 176 0.0824 8051 3 4217 20 60.3 0.3353 9673 5 5155 177 0.0821 3834 3 4238 20 60.4 0.3359 4827 5 5332 178 0.0817' 9596 3 4258 20 60.5 0.3365 0159 5 5510 179 0.0814 5338 3 4278 19 60.6 0.3370 5669 5 5688 179 0.0811 1060 3 4297 19 60.7 0.3376 1357 5 5868 180 0.0807 6763 3 4316 19 60.8 0.3381 7225 5 6048 181 0.0804 2446 3 4335 18 60.9 0.3387 3274 5 6229 182 0.0800 8111 3 4354 18
61.0 0.3392 9503 5 6412 183 0.0797 3758 3 4372 18 61.1 0.3398 5915 5 6595 184 0.0793 9386 3 4389 17 61.2 0.3404 2509 5 6778 185 0.0790 4997 3 4407 17 61.3 0.3409 9288 5 6963 186 0.0787 0590 3 4424 17 61.4 0.3415 6251 5 7149 187 0.0783 6167 3 4440 16
61,5 0.3421 3400 5 7336 188 0.0780 1727 3 4456 16 61.6 0.3427 0735 5 7523 188 0.0776 7270 3 4472 15 61.7 0.3432 8258 5 7712 189 0.0773 2798 3 4488 15 61.8 0.3438 5970 5 7901 190 0.0769 8310 3 4503 15 61.9 0.3444 3871 5 8091 191 0.0766 3807 3 4518 14
62.0 0.3450 1962 5 8283 192 0.0762 9290 3 4532 14 62.1 0.3456 0245 5 8475 193 0.0759 4758 3 4546 14 62.2 0.3461 8720 5 8668 194 0.0756 0212 3 4560 13 62.3 0.3467 7388 5 8863 195 0.0752 5652 3 4573 13 62.4 0.3473 6250 5 9058 196 1.0749 1079 3 4586 12
62.5 0.3479 5308 5 9254 197 0.0745 6494 3 4598 12 62.6 0.3485 4562 5 9451 198 0.0742 1895 3 4610 12 62.7 0.3491 4014 5 9650 199 0.0738 7285 3 4622 11 62.8 0.3497 3664 5 9849 200 0.0735 2664 3 4633 11 62.9 0.3503 3513 6 0050 202 0.0731 8030 3 4644 10
63.0 0.3509 3563 6 0251 203 0.0728 3387 3 4654 10 63.1 0.3515 3814 6 0454 204 0.0724 8732 3 4664 10 63.2 0.3521 4268 6 0658 205 0.0721 4068 3 4674 9 63.3 0.3527 4925 6 0862 206 0.0717 9394 3 4683 9 63.4 0.3533 5787 6 1068 207 0.0714 4711 3 4692 8
63.5 0.3539 6856 6 1275 2G8 0.0711 0019 3 4700 8 • 63.6 0.3545 8131 6 1483 209 0.0707 5319 3 4708 8 63.7 0.3551 9614 6 1693 210 0.0704 0610 3 4716 7 63.8 0.3558 1307 6 1903 212 0.0700 5895 3 4723 7 63.9 0.3564 3211 6 2115 213 0.0697 1172 3 4729 6
64.0 0.3570 5325 6 2328 214 0.0693 6442 3 4736 6 64.1 0.3576 7653 6 2542 215 0.0690 1706 3 4741 5 64.2 0.3583 0195 6 2757 216 0.0686 6965 3 4747 5 64.3 0.3589 2952 6 2974 218 0.0683 2218 3 4752 4 64.4 0.3595 5926 6 3191 219 0.0679 7466 3 4756 4
64.5 0.3601 9117 6 3410 220 0.0676 2710 3 4760 4 64.6 0.3608 2527 6 3630 221 0.0672 7950 3 4764 3 64.7 0.3614 6158 6 3852 223 0.0669 3186 3 4767 3 64.8 0.3621 0009 6 4075 224 0.0665 8420 3 4769 2 64.9 0.3627 4084 6 4299 225 0.0662 3650 3 4772 2 65.0 0.3633 8383 6 4524 227 0.0658 8879 3 4773 1 2o8 Bit lie tin of the Bureau of Standards [ Vol. 8, No. I
TABLE XIII— Continued
y LogF A: A 2 LogE Ai A2
65?0 0.3633 8383 6 4524 227 0.0658 8879 3 4773 1 65.1 0.3640 2907 6 4751 228 0.0655 4106 3 4774 1
65.2 0.3646 7658 6 4979 229 0.0651 9331 3 4775 +0 65.3 0.3653 2637 6 5209 231 0.0648 4556 3 4775 -0 65.4 0.3659 7846 6 5439 232 0.0644 9781 3 4775 1
65.5 0.3666 3286 6 5672 234 0.0641 5005 3 4775 1
65.6 0.3672 8957 6 5905 235 0.0638 0231 3 4773 2 65.7 0.3679 4863 6 6141 237 0.0634 5457 3 4772 2 65.8 0.3686 1003 6 6377 238 0.0631 0686 3 4769 3 65.9 0.3692 7380 6 6615 239 0.0627 5916 3 4767 3
66.0 0.3699 3995 6 6855 241 0.0624 1150 3 4764 4 66.1 0.3706 0850 6 7096 242 0.0620 6386 3 4760 4 66.2 0.3712 7946 6 7338 244 0.0617 1626 3 4756 5 66.3 0.3719 5284 6 7582 246 0.0613 6870 3 4751 5 66.4 0.3726 2866 6 7828 247 0.0610 2119 3 4746 6
66.5 0.3733 0694 6 8075 249 0.0606 7373 3 4740 6 66.6 0.3739 8768 6 8324 250 0.0603 2633 3 4734 7 66.7 0.3746 7092 6 8574 252 0.0599 7899 3 4727 7 66.8 0.3753 5666 6 8826 254 0.0596 3172 3 4720 8 66.9 0.3760 4492 6 9080 255 0.0592 8453 3 4712 8
67.0 0.3767 3572 6 9335 257 0.0589 3741 3 4703 9 67.1 0.3774 2907 6 9592 259 0.0585 9037 3 4695 9 67.2 0.3781 2499 6 9851 260 0.0582 4343 3 4685 10 67.3 0.3788 2349 7 0111 262 0.0578 9658 3 4675 11 67.4 0.3795 2460 7 0373 264 0.0575 4983 3 4664 "V, 67.5 0.3802 2833 7 0637 266 0.0572 0318 3 4653 12 / 67.6 0.3809 3471 7 0903 268 0.0568 5665 3 4642 12 67.7 0.3816 4373 7 1170 269 0.0565 1023 3 4629 13 67.8 0.3823 5544 7 1440 271 0.0561 6394 3 4617 13 67.9 0.3830 6984 7 1711 273 0.0558 1777 3 4603 14
68.0 0.3837 8695 7 1984 275 0.0554 7174 3 4589 15 68.1 0.3845 0679 7 2259 277 0.0551 2585 3 4575 15 68.2 0.3852 2938 7 2536 279 0.0547 8011 3 4559 16 68.3 0.3859 5475 7 2815 281 0.0544 3451 3 4544 16 68.4 0.3866 8290 7 3096 283 0.0540 8908 3 4527 17
68.5 0.3874 1386 7 3379 285 0.0537 4380 3 4510 18 68.6 0.3881 4765 7 3664 287 0.0533 9870 3 4493 18 68.7 0.3888 8429 7 3951 289 , 0.0530 5377 3 4475 19 68.8 0.3896 2380 7 4240 291 0.0527 0903 3 4456 19 68.9 0.3903 6620 7 4531 293 0.0523 6447 3 4436 20 69.0 0.3911 1152 7 4825 296 0.0520 2010 3 4416 21 69.1 0.3918 5977 7 5120 298 0.0516 7594 3 4396 21 69.2 0.3926 1097 7 5418 300 0.0513 3198 3 4375 22 69.3 0.3933 6515 7 5718 302 0.0509 8824 3 4353 23 69.4 0.3941 2234 7 6020 305 0.0506 4471 3 4330 23 69.5 0.3948 8254 7 6325 307 0.0503 0141 3 4307 24 69.6 0.3956 4579 7 6632 309 0.0499 5834 3 4283 24 69.7 0.3964 1211 7 6941 312 0.0496 1551 3 4259 25 69.8 0.3971 8152 7 7253 314 0.0492 7292 3 4233 26 69.9 0.3979 5405 7 7567 317 0.0489 3059 3 4208 26 70.0 0.3987 2972 7 7883 319 0.0485 8851 3 4181 27 Rosa I Formulas for Mutual and Self-Inductance Grover\ 209
TABLE XIII— Continued
y LogF Aj A 2 LogE A, A 2
70?0 0.3987 2972 7 7883 319 0.0485 8851 3 4181 27 70.1 0.3995 0855 7 8202 322 0.0482 4670 3 4154 28 70.2 0.4002 9058 7 8524 324 0.0479 0516 3 4126 29 70.3 0.4010 7582 7 8848 327 0.0475 6390 3 4098 29 70.4 0.4018 6430 7 9175 329 0.0472 2292 3 4068 30 70.5 0.4026 5605 7 9504 332 0.0468 8224 3 4039 31 70.6 0.4034 5109 7 9836 335 0.0465 4185 3 4008 31 70.7 0.4042 4945 8 0171 337 0.0462 0177 3 3977 32 70.8 0.4050 5116 8 0508 340 0.0458 6201 3 3945 33 70.9 0.4058 5625 8 0849 343 0.0455 2256 3 3912 33 71.0 0.4066 6474 8 1192 346 0.0451 8344 3 3879 34 71.1 0.4074 7666 8 1538 349 0.0448 4465 3 3844 35 71.2 0.4082 9204 8 1887 352 0.0445 0621 3 3810 36 71.3 0.4091 1090 8 2239 355 0.0441 6812 3 3774 36 71.4 0.4099 3329 8 2594 358 0.0438 3038 3 3738 37 71.5 0.4107 5923 8 2952 361 0.0434 9300 3 3700 38 71.6 0.4115 8875 8 3313 364 0.0431 5600 3 3663 39 71.7 0.4124 2187 8 3677 367 0.0428 1937 3 3624 39 71.8 0.4132 5864 8 4044 371 0.0424 8313 3 3585 40 71.9 0.4140 9909 8 4415 374 0.0421 4729 3 3544 41
72.0 0.4149 4324 8 4789 377 0.0418 1184 3 3504 42 72.1 0.4157 9112 8 5166 381 0.0414 7681 3 3462 42 72.2 0.4166 4279 8 5547 384 0.0411 4219 3 3419 43 72.3 0.4174 9826 8 5931 388 0.0408 0799 3 3376 44 72.4 0.4183 5757 8 6319 391 0.0404 7423 3 3332 45 72.5 0.4192 2076 8 6710 395 0.0401 4091 3 3287 46 72.6 0.4200 8786 8 7105 399 0.0398 0804 3 3241 46 72.7 0.4209 5891 8 7503 402 0.0394 7563 3 3195 47 72.8 0.4218 3394 8 7906 406 0.0391 4368 3 3148 48 72.9 0.4227 1300 8 8312 410 0.0388 1220 3 3099 49 73.0 0.4235 9612 8 8722 414 0.0384 8121 3 3050 50 73.1 0.4244 8334 8 9136 418 0.0381 5070 3 3001 51 73.2 0.4253 7470 8 9554 422 0.0378 2070 3 2950 52 73.3 0.4262 7023 8 9976 426 0.0374 9120 3 2898 52 73.4 0.4271 6999 9 0402 430 0.0371 6221 3 2846 53
73.5 0.4280 7401 9 0832 435 0.0368 3375 3 2793 54 73.6 0.4289 8233 9 1267 439 0.0365 0582 3 2739 55 73.7 0.4298 9499 9 1706 443 0.0361 7843 3 2684 56 73.8 0.4308 1205 9 2149 448 0.0358 5160 3 2628 57 73.9 0.4317 3354 9 2597 452 0.0355 2532 3 2571 58 74.0 0.4326 5950 9 3049 457 0.0351 9961 3 2513 59 74.1 0.4335 9000 9 3506 462 0.0348 7448 3 2455 60 74.2 0.4345 2506 9 3968 467 0.0345 4993 3 2395 60 74.3 0.4354 6474 9 4435 472 0.0342 2598 3 2335 61 74.4 0.4364 0909 9 4906 477 0.0339 0263 3 2273 62 74.5 0.4373 5815 9 5583 482 0.0335 7989 3 2211 63 74.6 0.4383 1198 9 5865 487 0.0332 5778 3 2148 64 74.7 0.4392 7063 9 6352 492 0.0329 3630 3 2084 65 74.8 0.4402 3414 9 6844 498 0.0326 1546 3 2019 66 74.9 0.4412 0258 9 7341 503 0.0322 9528 3 1952 67 75.0 0.4421 7599 9 7844 509 0.0319 7575 3 1885 68 2IO Bulletin of the Bureau of Standards [ Vol. 8, No. I
TABLE XIII— Continued
y LogF Ax A 2 LogE 1 A, A 2
75?0 0.4421 7599 9 7844 509 0.0319 7575 3 1885 68 75.1 0.4431 5444 9 8353 514 0.0316 5690 3 1817 69 75.2 0.4441 3797 9 8867 520 0.0313 3872 • 3 1748 70 75.3 0.4451 2664 9 9387 526 0.0310 2124 3 1678 71 75.4 0.4461 2051 9 9913 532 0.0307 0446 I 3 1607 72
75.5 0.4471 1965 10 0446 538 0.0303 8839 3 1535 73 75.6 0.4481 2410 10 0984 544 0.0300 7304 3 1462 74 75.7 0.4491 3394 10 1528 551 0.0297 5842 3 1388 75 75.8 0.4501 4922 10 2079 557 0.0294 4454 3 1313 76 75.9 0.4511 7001 10 2637 564 0.0291 3141 3 1237 77 76.0 0.4521 9638 10 3201 571 0.0288 1904 3 1159 78 76.1 0.4532 2839 10 3771 578 0.0285 0745 3 1081 79 76.2 0.4542 6610 10 4349 585 0.0281 9664 3 1002 80 76.3 0.4553 0959 10 4934 592 0.0278 8663 3 0921 82 76.4 0.4563 5893 10 5526 599 0.0275 7742 3 0839 83
76.5 0.4574 1419 10 6126 607 0.0272 6902 3 0757 84 76.6 0.4584 7545 10 6733 615 0.0269 6145 3 0673 85 76.7 0.4595 4278 10 7347 622 0.0266 5472 3 0588 86 76.8 0.4606 1625 10 7970 630 0.0263 4884 3 0502 87 76.9 0.4616 9594 10 8600 639 0.0260 4382 3 0415 88 77.0 0.4627 8195 10 9239 647 0.0257 3967 3 0327 89 77.1 0.4638 7433 10 9886 656 0.0254 3640 3 0237 91 77.2 0.4649 7319 11 0541 664 0.0251 3403 3 0147 92 77.3 0.4660 7860 11 1206 673 0.0248 3257 3 0055 93 77.4 0.4671 9066 11 1879 682 0.0245 3202 2 9962 94 77.5 0.4683 0945 11 2561 692 0.0242 3240 2 9868 95 77.6 0.4694 3506 11 3253 701 0.0239 3372 2 9772 97 77.7 0.4705 6760 11 3954 711 0.0236 3600 2 9676 98 77.8 0.4717 0714 11 4665 721 0.0233 3925 2 9578 99 77.9 0.4728 5379 11 5386 731 0.0230 4347 2 9479 100 78.0 0.4740 0766 11 6118 742 0.0227 4868 2 9378 102 78.1 0.4751 6884 11 6860 753 0.0224 5490 2 9277 103 78.2 0.4763 3743 11 7612 764 0.0221 6213 2 9174 104 78.3 0.4775 1355 11 8376 775 0.0218 7039 2 9070 105 78.4 0.4786 9731 11 9150 786 0.0215 7969 2 8964 107 78.5 0.4798 8881 11 9937 798 0.0212 9005 2 8858 108 78.6 0.4810 8818 12 0735 810 0.0210 0148 2 8750 109 78.7 0.4822 9553 12 1545 823 0.0207 1398 2 8640 111 78.8 0.4835 1098 12 2368 835 0.0204 2758 2 8529 112 78.9 0.4847 3466 12 3203 848 0.0201 4229 2 8417 113
79.0 0.4859 6669 12 4052 862 0.0198 5811 2 8304 115
79.1 0.4872 0721 12 4914 876 0.0195 7507 2 8189 116
79.2 0.4884 5635 12 5789 890 0.0192 9318 2 8073 118
79.3 0.4897 1424 12 6679 904 0.0190 1246 2 7955 119
79.4 0.4909 8103 12 7583 919 0.0187 3291 2 7836 120
79.5 0.4922 5687 12 8503 934 0.0184 5454 2 7716 122
79.6 0.4935 4189 12 9437 950 0.0181 7739 2 7594 123
79.7 0.4948 3626 13 0387 966 0.0179 0145 2 7470 125
79.8 0.4961 4013 13 1353 983 0.0176 2675 2 7345 126
79.9 0.4974 5367 13 2336 1000 0.0173 5330 2 7219 128
80.0 0.4987 7703 13 3336 1018 0.0170 8111 2 7091 129
Rosa
Grover
~J Formulas for Mutual and Self-Inductance 211
TABLE XIII— Continued
V LogF Ai A 2 LogE A, A 2
80?0 0.4987 7703 13 3336 1018 0.0170 8111 2 7091 129 80.1 0.5001 1040 13 4354 1036 0.0168 1020 2 6962 131 80.2 0.5014 5394 13 5390 1054 0.0165 4058 2 6831 132 80.3 0.5028 0783 13 6444 1073 0.0162 7227 2 6698 134 80.4 0.5041 7227 13 7517 1093 0.0160 0529 2 6564 136
80.5 0.5055 4744 13 8610 1113 0.0157 3965 2 6429 137 80.6 0.5069 3354 13 9724 1134 0.0154 7536 2 6291 139 80.7 0.5083 3078 14 0858 1156 0.0152 1245 2 6153 140 80.8 0.5097 3936 14 2014 1178 0.0149 5092 2 6012 142 80.9 0.5111 5949 14 3192 1201 0.0146 9080 2 5870 144
81.0 0.5125 9141 14 4393 1225 0.0144 3210 2 5726 145 81.1 0.5140 3534 14 5617 1249 0.0141 7484 2 5581 147 81.2 0.5154 9151 14 6867 1274 0.0139 1903 2 5433 149 81.3 0.5169 6018 14 8141 1300 0.0136 6470 2 5285 151 81.4 0.5184 4159 14 9441 1327 0.0134 1185 2 5134 152
81.5 0.5199 3600 . 15 0769 1355 0.0131 6052 2 4981 154 81.6 0.5214 4369 15 2124 1384 0.0129 1070 2 4827 156 81.7 0.5229 6493 15 3508 1414 0.0126 6243 2 4671 158 81.8 0.5245 0001 15 4922 1445 0.0124 1572 2 4513 160 81.9 0.5260 4923 15 6366 1477 0.0121 7058 2 4354 162
82.0 0.5276 1289 15 7843 1510 0.0119 2704 2 4192 163 82.1 0.5291 9132 15 9352 1544 0.0116 8512 2 4029 165 82.2 0.5307 8485 16 0896 1579 0.0114 4483 2 3863 167 82.3 0.5323 9381 16 2476 1616 0.0112 0620 2 3696 169 82.4 0.5340 1857 16 4092 1655 0.0109 6924 2 3527 171
82.5 0.5356 5949 16 5747 1694 0.0107 3397 2 3356 173 82.6 0.5373 1696 16 7441 1736 0.0105 0041 2 3183 175 82.7 0.5389 9137 16 9177 1779 0.0102 6859 2 3007 177 82.8 0.5406 8313 17 0955 1823 0.0100 3851 2 2830 179 82.9 0.5423 9268 17 2778 1870 0.0098 1021 2 2651 181
83.0 0.5441 2047 17 4648 1918 0.0095 8371 2 2469 184 83.1 0.5458 6695 17 6566 1968 0.0093 5902 2 2285 186 83.2 0.5476 3260 17 8534 2021 0.0091 3616 2 2100 188 83.3 0.5494 1795 18 0555 2076 0.0089 1517 2 1912 190 83.4 0.5512 2350 18 2631 2133 0.0086 9605 2 1721 193
83.5 0.5530 4980 18 4764 2193 0.0084 7884 2 1529 195 83.6 0.5548 9744 18 6956 2255 0.0082 6355 2 1334 197 83.7 0.5567 6700 18 9211 2320 0.0080 5021 2 1137 199 83.8 0.5586 5912 19 1532 2389 0.0078 3884 2 0937 202 83.9 0.5605 7443 19 3921 2460 0.0076 2947 2 0735 204 84.0 0.5625 1364 19 6381 2535 0.0074 2211 2 0531 207 84.1 0.5644 7745 19 8916 2614 0.0072 1680 2 0324 209 84.2 0.5664 6661 20 1531 2697 0.0070 1356 2 0115 212 84.3 0.5684 8192 20 4228 2784 0.0068 1241 1 9903 214 84.4 0.5705 2420 20 7012 2875 0.0066 1338 1 9689 217 84.5 0.5725 9431 20 9887 2972 0.0064 1649 1 9472 220 84.6 0.5746 9318 21 2859 3073 0.0062 2177 1 9252 222 84.7 0.5768 2177 21 5932 3180 0.0060 2925 1 9029 225 84.8 0.5789 8109 21 9112 3293 0.0058 3896 1 8804 228 84.9 0.5811 7221 22 2405 3413 0.0056 5092 1 8576 231 85.0 0.5833 9626 22 5818 3539 0.0054 6516 1 8345 234 212 Bulletin of the Bureau of Standards [Vol.8. .Vo. i
TABLE XIII— Continued
Y LogF A, A 2 LogE Aj A 2
85?0 0.5833 9626 22 5818 3539 0.0054 6516 1 8345 234 85.1 0.5856 5444 22 9357 3673 0.0052 8171 1 8111 237 85.2 0.5879 4801 23 3031 3816 0.0051 0060 1 7874 240 85.3 0.5902 7832 23 6846 3967 0.0049 2185 1 7634 243 85.4 0.5926 4679 24 0813 4127 0.0047 4551 1 7391 246 85.5 0.5950 5492 24 4940 4299 0.0045 7160 1 7145 249 85.6 0.5975 0432 24 9239 4481 0.0044 0015 1 6896 253 85.7 0.5999 9671 25 3720 4676 0.0042 3119 1 6643 256 85.8 0.6025 3391 25 8396 4885 0.0040 6476 1 6387 260 85.9 0.6051 1788 26 3281 5109 0.0039 0089 1 6127 263 86.0 0.6077 5069 26 8390 5349 0.0037 3962 1 5864 267 86.1 0.6104 3459 27 3739 5607 0.0035 8097 1 5598 270 86.2 0.6131 7198 27 9346 5886 0.0034 2499 1 5327 274 86.3 0.6159 6543 28 5231 6186 0.0032 7172 1 5053 278 86.4 0.6188 1775 29 1418 6512 0.0031 2118 1 4775 282 86.5 0.6217 3193 29 7929 6865 0.0029 7343 1 4493 286 86.6 0.6247 1122 30 4794 7248 0.0028 2850 1 4207 290 86.7 0.6277 5916 31 2042 7667 0.0026 8642 1 3917 295 86.8 0.6308 7958 31 9709 8124 0.0025 4725 1 3622 299 86.9 0.6340 7668 32 7834 8626 0.0024 1103 1 3323 304 87.0 0.6373 5501 33 6459 9177 0.0022 7779 1 3020 308 87.1 0.6407 1961 34 5636 9785 0.0021 4759 1 2712 313 87.2 0.6441 7597 35 5422 10459 0.0020 2048 1 2398 318 87.3 0.6477 3019 36 5881 11208 0.0018 9649 1 2080 324 87.4 0.6513 8900 37 7089 12043 0.0017 7569 1 1757 329 87.5 0.6551 5989 38 9132 12980 0.0016 5813 1 1428 335 87.6 0.6590 5121 40 2112 14035 0.0015 4385 1 1093 340 87.7 0.6630 7233 41 6147 15230 0.0014 3292 1 0753 347 87.8 0.6672 3380 43 1377 16590 0.0013 2540 1 0406 353 87.9 0.6715 4757 44 7967 18149 0.0012 2134 1 0053 360 88.0 0.6760 2724 46 6116 19948 0.0011 2081 9693 367 88.1 0.6806 8840 48 6064 22040 0.0010 2387 9327 374 88.2 0.6855 4904 50 8104 24492 0.0009 3060 8953 382 88.3 0.6906 3009 53 2597 27396 0.0008 4107 8571 390 88.4 0.6959 5605 55 9993 30870 0.0007 5536 8181 399 88.5 0.7015 5598 59 0862 35077 0.0006 7355 7782 408 88.6 0.7074 6460 62 5940 40245 0.0005 9573 7374 418 88.7 0.7137 2400 66 6184 46693 0.0005 2199 6956 429 88.8 0.7203 8584 71 2878 54895 0.0004 5242 6527 441 88.9 0.7275 1462 76 7773 65561 0.0003 8715 6087 453 89.0 0.7351 9234 83 3334 79812 0.0003 2628 5633 467 89.1 0.7435 2568 91 3146 99496 0.0002 6995 5166 483 89.2 0.7526 5714 101 2642 127847 0.0002 1829 4683 501 89.3 0.7627 8356 114 0489 170975 0.0001 7146 4181 522 89.4 0.7741 8844 131 1464 241655 0.0001 2965 3660 546 89.5 0.7873 0308 155 3119 370693 0.0000 9305 3114 576 89.6 0.8028 3427 192 3813 650756 0.0000 6192 2538 615 89.7 0.8220 7240 257 4569 1501510 0.0000 3654 1923 670 89.8 0.8478 1809 407 6079 0.0000 1731 1253 774 89.9 0.8885 7889 0.0000 0479 479 90.0 Inf. 0.0000 0000 Rosa Formulas for Mutual and Self-Inductance "1
Grover\ 213
The preceding table of logarithms of the elliptic integrals of the first and second kinds is taken from L,egendre's Traitk des Fonctions Elliptiques, volume 2, Table I. The values from 45 ° to 90 ° are given for intervals of o?i. The values from o° to 45 °, which are com- paratively seldom required, have been omitted. For formula and table to be used in interpolation, see page 214.
TABLE XIV
Binominal Coefficients for Interpolation by Differences
Coefficients of Coefficients of Coefficients of Coefficients of
A 2 and A3 A2 and A 3 A 2 and A 3 A 2 and A3
k k k k K 2 K3 K 2 K3 K 2 K3 K 2 K 3
0.01 -0.005 +0.003 0.26 -0.096 +0.056 0.51 -0.125 4-0.062 0.76 -0.091 +0.038 .02 - .010 + .006 .27 - .099 + .057 .52 - .125 + .062 .77 - .089 + .036 .03 - .015 + .010 .28 - .101 + .058 .53 - .125 + .061 .78 - .086 + .035 .04 - .019 + .013 .29 - .103 + .059 .54 - .124 + .060 .79 - .083 + .033 .05 - .024 + .015 .30 - .105 + .060 .55 - .124 + .060 .80 - .080 + .032 .06 - .028 + .018 .31 - .107 + .060 .56 - .124 + .059 .81 - .077 + .031 .07 - .033 + .021 .32 - .109 + .061 .57 - .123 + .058 .82 - .074 + .029 .08 - .037 + .024 .33 - .111 + .062 .58 - .122 + .058 .83 - .071 + .028 .09 - .041 + .026 .34 - .112 + .062 .59 - .121 + .057 .84 - .067 + .026 .10 - .045 + .028 .35 - .114 + .063 .60 - .120 + .056 .85 - .064 + .024 .11 - .049 + .031 .36 - .115 + .063 .61 - .119 + .055 .86 - .060 + .023 .12 - .053 4- .033 .37 - .117 + .063 .62 - .118 + .054 .87 - .057 + .021 .13 - .057 + .035 .38 - .118 + .064 .63 - .117 + .053 .88 - .053 + .020 .14 - .060 + .037 .39 - .119 + .064 .64 - .115 + .052 .89 - .049 + .018 .15 - .064 + .039 .40 - .120 + .064 .65 - .114 + .051 .90 - .045 + .016 .16 - .067 + .041 .41 - .121 + .064 .66 - .112 + .050 .91 - .041 + .015 .17 - .071 + .043 .42 - =122 + .064 .67 - .111 + .049 .92 - .037 + .013 .18 - .074 + .045 .43 - .123 + .064 .68 - .109 + .048 .93 - .033 + .012 .19 - .077 + .046 .44 - .123 + .064 .69 - .107 + .047 .94 - .028 + .010 .20 - .080 + .048 .45 - .124 + .064 .70 - .105 + .045 .95 - .024 + .008 .21 - .083 + .049 .46 - .124 + .064 .71 - .103 + .044 .96 - .019 + .007 .22 - .086 + .051 .47 - .125 + .064 .72 - .101 + .043 .97 - .015 + .005 .23 - .089 + .052 .48 - .125 -I- .063 .73 - .099 + .042 .98 - .010 + .003 .24 - .091 + .053 .49 - .125 + .063 .74 - .096 + .040 .99 - .005 + .002 .25 - .094 + .055 .50 - .125 + .063 .75 - .094 + .039 1.00 - .000 + .000 2
214 Bulletin of the Bureau of Standards Woi. 8, No. i
INTERPOLATION FORMULA
*(*-i)(*-. a )(*- 3 ) ..
4 !
or,/(fl + A)=/(a)+^A 1 + A'g A + ^sA + 8 8 (o)
where the constants K and K are given in the above table as % z
functions of k and h k~ where h is the remainder above the value of a for which the func- tion is given in the table, and 8 is the increment of a in the table.
ILLUSTRATION
To find the value of log F for 49 15' 36" = 49?26o For 49? log ^=0.2836 3i3o=y(#) ^ = .06, 8 = 0.1 £ = 0.6 From Table XIV, A"2 = - . 1 20 K,= +.056 From Table XIII, A = 39338 x
A = 117
2
A =
3
1
Substituting these values of K^ K^ A A A 1} 8, 3 in formula (£) above we have as the value of log F for the given angle log ^=0.2836 3130 + 0.0002 3603 — 0.00000014 = 0.2838 6719. GrTver] Formulas for Mutual and Self-Inductance 215
TABLE XV
Values of the Quantities q—? or Q \~ ~k and ^°<?io (2+e) with Argument q or q x
e=3q 4 -44 +9q 8 -i24 10 +
(i k
* *
(For use with Formulas (8), (9), (45), (76), (77), and (78))
q
orqi H A e A Logio (1+e) A «*-!
0.020 0.000 00001 0.000 00048 22 0.000 00021 9 .022 .000 00001 1 .000 00070 29 .000 00030 13 .024 .000 00002 .000 00099 38 .000 00043 16 .026 .000 00002 1 .000 00137 47 .000 00059 21 .028 .000 00003 2 .000 00184 59 .000 00080 25
0.030 0.000 00005 2 0.000 00243 71 0.000 00105 31 .032 .000 00007 2 .000 00314 86 .000 00136 38 .034 .000 00009 3 .000 00400 103 .000 00174 44 .036 .000 00012 4 .000 00503 121 .000 00218 53 .038 .000 00016 5 .000 00624 142 .000 00271 61
0.040 0.000 00021 5 0.000 00766 165 0.000 00332 72 .042 .000 00026 7 .000 00931 191 .000 00404 83 .044 .000 00033 8 .000 01122 217 .000 00487 94 .046 .000 00041 10 .000 01339 249 .000 00581 109 .048 .000 00051 12 .000 01588 280 .000 00690 122
0.050 0.000 00063 13 0.000 01868 318 0.000 00812 138 .052 .000 00076 16 .000 02186 355 .000 00950 154 .054 .000 00092 18 .000 02541 397 .000 01104 172 .056 .000 00110 21 .000 02938 442 .000 01276 192 .058 .000 00131 25 .000 03380 490 .000 01468 213
0.060 0.000 00156 27 0.000 03870 540 0.000 01681 234 .062 .000 00183 32 .000 04410 596 .000 01915 259 .064 .000 00215 36 .000 05006 654 .000 02174 283 .066 .000 00251 40 .000 05660 715 .000 02457 312 .068 .000 00291 45 .000 06375 781 .000 02769 339
0.070 0.000 00336 51 0.000 07156 851 0.000 03108 369 .072 .000 00387 57 .000 08007 924 .000 03477 401 .074 .000 00444 63 .000 08931 1002 .000 03878 436 .076 .000 00507 70 .000 09933 1083 .000 04314 470 .078 .000 00577 78 .000 11016 1169 .000 04784 509 2l6 Bulletin of the Bureau of Standards Woi. s, No. i
TABLE XV— Continued
q or qi H «*-! A s A Logio (1+e) A
0.080 0.000 00655 86 0.000 12185 1259 0.000 05293 545 .082 .000 00741 95 .000 13444 1354 .000 05838 588 .084 .000 00836 105 .000 14798 1453 .000 06426 631 .086 .000 00941 114 ! .000 16251 1557 .000 07057 676 .088 .000 01055 126 i .000 17808 1666 .000 07733 724
0.090 0.000 01181 137 0.000 19474 1779 0.000 08457 772 .092 .000 01318 150 i
.000 21253 1899 .000 09229 825
.094 .000 01468 162 .000 23152 2022 .000 10054 878 .096 .000 01630 177 .000 25174 2150 .000 10932 937 .098 .000 01807 193 .000 27324 2285 .000 11869 988
0.100 0.000 02000 102 0.000 29609 1194 0.000 12857 519 .101 .000 02102 106 .000 30803 1230 .000 13376 533 .102 .000 02208 110 .000 32033 1266 .000 13909 550 .103 .000 02318 115 ! .000 33299 1303 .000 14459 566 .104 .000 02433 119 1 .000 34602 1340 .000 15025 582
0.105 0.000 02552 123 ;
0.000 35942 1379 0.000 15607 598
.106 .000 02675 129 .000 37321 1410 .000 16205 616 .107 .000 02804 134 .000 38731 1465 .000 16821 632 .108 .000 02938 138 .000 40196 1498 .000 17453 651 .109 .000 03076 144 .000 41694 1539 .000 18104 668
0.110 0.000 03220 149 0.000 43233 1581 0.000 18772 686 .111 .000 03369 154 .000 44814 1624 .000 19458 705 .112 .000 03523 160 .000 46438 1667 .000 20163 724 .113 .000 03683 166 .000 48105 1711 .000 20887 742 .114 .000 03849 172 .000 49816 1756 .000 21629 762
0.115 0.000 04021 178 0.000 51572 1802 0.000 22391 783 .116 .000 04199 184 .000 53374 1848 .000 23174 802 .117 .000 04383 191 .000 55222 1895 .000 23976 823 .118 .000 04574 196 .000 57117 1943 .000 24799 843 .119 .000 04770 204 .000 59060 1992 .000 25642 865
0.120 0.000 04974 210 0.000 61052 2041 0.000 26507 885 .121 .000 05184 218 .000 63093 2091 .000 27392 908 .122 .000 05402 226 .000 65184 2143 .000 28300 930 .123 .000 05628 232 .000 67327 2195 .000 29230 953 .124 .000 05860 240 .000 69522 2247 .000 30183 975
0.125 0.000 06100 248 0.000 71769 2301 0.000 31158 998 .126 .000 06348 255 .000 74070 2355 .000 32156 1022 .127 .000 06603 265 .000 76425 2410 .000 33178 1046 .128 .000 06868 272 .000 78835 2466 .000 34224 1071 .129 .000 07140 280 .000 81301 2523 .000 35295 1094 Rosa GroverS "J Formulas for Mutual and Self-Inductance 217
TABLE XV— Continued
Q
or qi H «*-& A s A Logio (1+e) A
0.130 0.000 07420 '290 0.000 83824 2581 0.000 36389 1120 .131 .000 07710 299 .000 86405 2639 .000 37509 1145 .132 .000 08009 308 .000 89044 2698 .000 38654 1171 .133 .000 08317 317 .000 91742 2759 .000 39825 1196 .134 .000 08634 327 .000 94501 2820 .000 41021 1224
0.135 0.000 08961 336 0.000 97321 2881 0.000 42245 1251 .136 .000 09297 347 .001 00202 2945 .000 43496 1277 .137 .000 09644 357 .001 03147 3012 .000 44773 1305 .138 .000 10001 367 .001 06155 3073 .000 46078 1333 .139 .000 10368 378 .001 09228 3138 .000 47411 1362
0.140 0.000 10746 389 0.001 12366 3204 0.000 48773 1389 .141 .000 11135 401 .001 15570 3272 .000 50162 1420 .142 .000 11536 411 .001 18842 3339 .000 51582 1448 .143 .000 11947 423 .001 22181 3409 .000 53030 1479 .144 .000 12370 435 .001 25590 3479 .000 54509 1509
0.145 0.000 12805 448 0.001 29069 3549 0.000 56018 1539 .146 .000 13253 459 .001 32618 3621 .000 57557 1571 .147 .000 13712 473 .001 36239 3694 .000 59128 1602 .148 .000 14185 485 .001 39933 3768 .000 60730 1634 .149 .000 14670 498 .001 43701 3842 .000 62364 1666
0.150 0.000 15168 0.001 47543 0.000 64030
Tables XV and XVI are reproduced from Nagaoka's paper; see footnote, page 12. 218 Bulletin of the Bureau of Standards [Vol. 8, No. 1
TABLE XVI
Values of and — e/ with Argument q
e2 x
61= 3 2^ -4o^ +48^ -
1
3 4 5
• • •
(For use with Formulas (9) and (9a))
Qi «l A — ei' A
0.0100 0.000 03160 93 0.000 76840 1499 .0099 .000 03067 92 .000 75341 1484 .0098 .000 02975 89 .000 73857 1471 .0097 .000 02886 89 .000 72386 1455 .0096 .000 02797 86 .000 70931 1442
0.0095 0.000 02711 84 0.000 69489 1428 .0094 .000 02627 83 .000 68061 1413 .0093 .000 02544 81 .000 66648 1399 .0092 .000 02463 78 .000 65249 1386 .0091 .000 02385 78 .000 63863 1370
0.0090 0.000 02307 76 0.000 62493 1356 .0089 .000 02231 74 .000 61137 1342 .0088 .000 02157 73 .000 59795 1327 .0087 .000 02084 71 .000 58468 1313 .0086 .000 02013 69 .000 57155 1299
0.0085 0.000 01944 67 0.000 55856 1285 .0084 .000 01877 67 .000 54571 1269 .0083 .000 01810 64 .000 53302 1256 .0082 .000 01746 62 .000 52046 1242 .0081 .000 01684 62 .000 50804 1226
0.0080 0.000 01622 60 0.000 49578 1212 .0079 .000 01562 59 .000 48366 1197 .0078 .000 01503 56 .000 47169 1184 .0077 .000 01447 55 .000 45985 1169 .0076 .000 01392 55 .000 44816 1153
0.0075 0.000 01337 52 0.000 43663 1140 .0074 .000 01285 51 .000 42523 1125 .0073 .000 01234 51 .000 41398 1109 .0072 .000 01183 48 .000 40289 1096 .0071 .000 01135 47 .000 39193 1081
0.0070 0.000 01088 46 0.000 38112 1066 .0069 .000 01042 45 .000 37046 1051 .0068 .000 00997 43 .000 35995 1037 .0067 .000 00954 42 .000 34958 1022 .0066 .000 00912 . 40 .000 33936 1008
0.0065 0.000 00872 40 0.000 32928 992 .0064 .000 00832 38 .000 31936 978 .0063 .000 00794 37 .000 30958 963 .0062 .000 00757 37 .000 29995 947 .0061 .000 00720 34 .000 29048 934 Rosa "I Formulas for Mutual and Self-Inductance Grover\ 219
TABLE XVI— Continued
qi ei A- -e,' A
0.0060 0.000 00686 34 0.000 28114 918 .0059 .000 00652 33 .000 27196 903 .0058 .000 00619 30 .000 26293 890 .0057 .000 00589 31 .000 25403 873 .0056 .000 00558 30 .000 24530 858
0.0055 0.000 00528 27 0.000 23672 845 .0054 .000 00501 28 .000 22827 828 .0053 .000 00473 26 .000 21999 814 .0052 .000 00447 26 .000 21185 798 .0051 .000 00421 24 .000 20387 784
0.0050 0.000 00397 23 0.000 19603 769 .0049 .000 00374 22 .000 18834 754 .0048 .000 00352 22 .000 18080 738 .0047 .000 00330 21 .000 17342 723 .0046 .000 00309 19 .000 16619 709
0.0045 0.000 00290 18 0.000 15910 694 .0044 .000 00272 19 .000 15216 677 .0043 .000 00253 17 .000 14539 663 .0042 .000 00236 16 .000 13876 648 .0041 .000 00220 16 .000 13228 632
0.0040 0.000 00204 15 0.000 12596 617 .0039 .000 00189 14 .000 11979 602 .0038 .000 00175 14 .000 11377 586 .0037 .000 00161 13 .000 10791 571 .0036 .000 00148 12 .000 10220 556
0.0035 0.000 00136 11 0.000 09664 541 .0034 .000 00125 11 .000 09123 525 .0033 .000 00114 9 .000 08598 511 .0032 .000 00105 10 .000 08087 494 .0031 .000 00095 9 .000 07593 479
0.0030 0.000 00086 8 0.000 07114 464 .0029 .000 00078 8 .000 06650 448 .0028 .000 00070 7 .000 06202 433 .0027 .000 00063 7 .000 05769 417 .0026 .000 00056 6 .000 05352 402
0.0025 0.000 00050 6 0.000 04950 386 .0024 .000 00044 5 .000 04564 371 .0023 .000 00039 5 .000 04193 355 .0022 .000 00034 4 .000 03838 340 .0021 .000 00030 4 .000 03498 324
0.0020 0.000 00026 4 0.000 03174 308 .0019 .000 00022 3 .000 02866 293 .0018 .000 00019 3 .000 02573 277 .0017 .000 00016 3 .000 02296 261 .0016 .000 00013 2 .000 02035 246 220 Bulletin of the Bureau of Standards [ Vol. 8, No. I
TABLE XVI— Continued
qi ei A — ei' A
0.0015 0.000 00011 2 0.000 01789 230
.0014 .000 00009 2 .000 01559 214
.0013 .000 00007 1 .000 01345 199
.0012 .000 00006 2 .000 01146 182
.0011 .000 00004 1 .000 00964 167
0.0010 0.000 00003 1 0.000 00797 151
.0009 .000 00002 .000 00646 136
.0008 .000 00002 1 .000 00510 119
.0007 .000 00001 .000 00391 104
.0006 .000 00001 1 .000 00287 87
0.0005 0.000 00000 0.000 00200 72
.0004 .000 00000 .000 00128 56
.0003 .000 00000 .000 00072 40
.0002 .000 00000 .000 00032 24
.0001 .000 00000 .000 00008
.
TABLE XVII
Coefficients of the Hypergeometric Series in Formula (18)
Series at a2 a3
/l 5 1 J-l\ 0.069 4444 0.035 5260 0.023 8485
*\l2' 12' 2' J /
-0.097 2222 -0.047 0358 -0.031 0523
V"!^ 12* 2* J J /5 13 3 J-l\ 0.300 9259 0.177 6300 0.126 0562 * V12' 12* 2' ] J
F /7 11 3 J-l
0.356 4814 0.216 3645 0.155 2615
*t,12' 12' 2' TV
Grfver] Formulas for Mutual and Self-Inductance 221
TABLE XVIII
Showing the Location and Magnitude of the Positive and Negative Maxima and the Positions of the Roots of the Coefficients in Gray's and Searle and Airey's Formulas
(For use in Formulas (40), (43), and (56))
X X X X
A x 2
A x A x 6
A x 8
3.0000 2.5000 2.1875 1.9688
0.8660 0.531 0.3898 0.3000 00 — 00 1.118 -3.750 0.6961 - 1.8273 0.5162 - 1.2177 1.489 0.9203 0.687 00 + 00 1.737 +30.69 1.1521 + 7.364 2.063 1.268 00 — 00 2.309 -570.97 2.613 00 + 00 X X X A Xio A Xi2 _ A XH
1.8407 1.6758 1.5710
0.2575 0.2193 0.1936
0.4145 - 0.924 0.3466 - 0.756 0.2992 - 0.6446
0.5520 0.4629 0.4010
0.8006 + 3.428 0.6475 + 2.000 0.5460 + 1.396
0.9386 0.7627 0.6439
1.413 -60.80 1.052 - 18.20 0.8515 - 8.166
1.589 1.145 0.9559
2.862 + 18892 1.734 +836.1 1.289 + 154.6
3.158 1.902 1.414
00 — 00 3.406 -963500 2.044 -16993 y
3.618 2.207
00 + 00 3.950 +70850000
4.226
00 — 00
The function X2n has n roots, between which values it makes oscil- lations of ever-increasing amplitude, and for values of -^ greater than
the largest root the function increases rapidly without limit. The functions L2n have the same form as X 2n) - being the variable instead of A 21674 — 12- •15 222 Bulletin of the Bureau of Standards [ Vol. 8, No. i
TABLE XIX
Values of Coefficients in Gray's and Searle and Airey's Formulas
(For use in Formulas (40) (43), and (56))
X A x 2 x 4 x 6 Xs Xio x 12 x„
0.0 + 3.000 + 2.500 + 2.188 + 1.969 + 1.841 + 1.676 + 1.571
0.1 2.960 2.400 2.015 1.712 1.494 1.234 + 1.032
0.2 2.840 2.106 1.521 1.017 + 0.618 + 0.216 - 0.073 0.3 2.640 1.632 + 0.780 + 0.090 - 0.355 - 0.635 - 0.645 0.4 2.360 1.002 - 0.090 - 0.764 - 0.874 - 0.596 - 0.0093
0.5 2.000 + 0.250 - 0.938 - 1.203 - 0.526 + 0.483 + 1.208 0.6 1.560 - 0.580 - 1.577 - 0.909 + 0.760 + 1.793 + 1.000 0.7 1.040 - 1.438 - 1.814 + 0.228 2.467 + 1.662 - 3.175 0.8 + 0.440 2.262 - 1.452 + 2.207 3.423 - 1.733 - 7.231 0.9 - 0.240 2.976 - 0.335 + 4.606 + 1.924 - 8.748 - 6.811
1.0 - 1.000 3.500 + 1.688 + 6.719 - 3.878 - 16.46 + 10.48 1.1 - 1.840 3.750 4.673 7.240 1.2 - 2.760 3.606 8.589 + 4.509 - 31.72 + 22.27 +119.6 1.3 - 3.760 _ 2.976 13.28 - 3.595 1.4 - 4.840 - 1.734 18.44 - 19.49 - 60.66 + 29.1 1.5 - 6.000 + 0.250 + 23.56 - 46.24 - 48.96 +765.7 -486.5 1.6 - 7.240 + 3.114 23.90 - 89.42 1.7 - 8.560 7.008 30.46 - 137.4 + 151.0 +818.1 1.8 - 9.960 12.09 29.83 - 205.2 1.9 -11.44 18.53 24.50 - 285.9 + 21.8 2.0 -13.00 + 26.50 + 12.19 - 375.0 + 1591 - 1969 -16740 2.1 -14.64 36.55 - 9.64 - 464.9 2.2 -16.36 47.80 - 44.09 - 538.3 4059 -14090 -1840 2.3 -18.16 62.54 - 104.9 - 570.9 2.4 -20.04" 77.61 - 166.3 - 535.5 2.5 -22.00 + 96.24 - 263.4 - 386.7 10908 -80050 2.6 -24.04 117.6 - 390.4 - 64.4 + 658400 2.7 -26.16 142.2 - 559.0 + 505.9 2.8 -28.36 169.0 - 801.2 1433 18390 -222400 2.9 -30.64 201.3 -1039.1 2833
3.0 -33.00 + 236.5 -1370.3 +4869 + 15797 -509200 19132000 3.25 -38.25 + 343.1 -2553 + 14118 3.5 -46,00 + 480.2 -4414 +33030 -146970 -893400 33670000 3.75 -53.25 + 653.1 -7215 +68410 +265600 4.0 -61.00 + 866.5 -11286 + 130400 -1.229X10 6 +6.625X10 6 59080000
4.25 -69.25 -16530000 4.5 -77.00 + 1440.3 -24956 +399000 -5.683X10 5 +5.972X10 7 -4.172X108 5.0 -97.00 +2252.5 -49810 + 1038700 -2.007X10' +3.463 XIO8 -4.855X109
This table used in conjunction with the preceding should make it possibleto investigate the convergence of Gray's or Searle and Airey's formula in any given case. It will also facilitate calculations
by these formulas when -^ has one of the values included in the A I table. This table gives also the values of the L2n coefficients if — be taken as argument in place of— • Rosa "1 Groverj Formulas for Mutual and Self-Inductance 223
TABLE XX
Nagaoka's Table of Values of the Correction Factor for the Ends K, as a Function of the -1 2a Angle Q=tan
(For use in Formula (75))
e K A! A 2
6 K A i
A 2
0° 1.000 000 - 7370 + 72 45° 0.688 423 - 7659 - 95
1 0.992 630 - 7298 67 46 .680 764 - 7754 - 102
2 .985 332 - 7231 63 47 .673 010 - 7856 - 108
3 .978 101 - .7168 60 48 .665 154 - 7964 - 115
4 .970 933 - 7109 56 49 .657 190 - 8079 - 120
5 0.963 825 - 7053 + 52 ' 50 0.649 111 - 8199 - 128
6 .956 771 - 7001 47 51 .640 912 - 8327 - 136
7 .949 770 - 6955 44 52 .632 585 - 8463 - 142
8 .942 815 - 6910 40 53 .624 122 - 8605 - 152
9 .935 906 - 6870 37 54 .615 517 - 8757 - 160
10 0.929 036 - 6833 + 34 55 0.606 760 - 8917 - 169. 11 .922 203 - 6799 30 56 .597 843 - 9086 - 179 12 .915 404 - 6769 27 57 .588 757 - 9265 - 190 13 .908 635 - 6742 24 58 .579 492 - 9455 - 200 14 .901 893 - 6718 19 59 .570 037 - 9655 - 214 15 0.895 175 - 6699 + 18 60 0.560 382 - 9869 - 226 16 .888 476 - 6681 14 61 .550 513 - 10095 - 239 17 .881 795 - 6667 10 62 .540 418 -10334 - 256 18 .875 128 - 6657 8 63 .530 084 -10590 - 270 19 .868 471 - 6649 4 64 .519 494 -10860 - 288 20 0.861 822 - 6645 + 2 65 0.508 634 -11148 - 308 21 .855 177 - 6643 - 2 66 .497 486 -11456 - 328 22 .848 534 - 6645 - 5 67 .486 030 -11784 - 351 23 .841 889 - 6650 - 9 68 .474 246 -12135 - 376 24 .835 239 - 6659 - 10 69 .462 111 -12511 - 403 25 0.828 580 - 6669 - 16 70 0.449 600 -12914 - 435 26 .821 911 - 6685 - 17 71 .436 686 -13349 - 467 27 .815 226 - 6702 - 21 72 .423 337 -13816 - 506 28 .808 524 - 6723 - 24 72 .409 521 -14322 - 549 29 .801 801 - 6747 - 28 74 .395 199 -14871 - 597 30 0.795 054 - 6775 - 32 75 0.380 328 -15468 - 653 31 .788 279 - 6807 - 34 76 .364 860 -16121 - 717 32 .781 472 - 6841 - 39 77 .348 739 - 16838 - 791 33 .774 631 - 6880 - 41 78 .331 901 -17629 - 881 34 .767 751 - 6921 - 46 79 .314 272 -18510 - 985 35 0.760 830 - 6967 - 50 80 0.295 762 -19495 - 1116 36 .753 863 - 7017 - 54 81 .276 267 -20611 - 1275 37 .746 846 - 7071 - 57 82 .255 656 -21886 - 1484 38 .739 775 - 7128 - 61 83 .233 770 -23370 - 1758 39 .732 647 - 7189 - 67 84 .210 400 -25128 - 2144 40 0.725 458 - 7256 - 71 85 0.185 272 -27272 - 2725 41 .718 202 - 7327 - 75 86 .158 000 -29997 - 3707 42 .710 875 - 7402 - 81 87 .128 003 -33704 - 5760 43 .703 473 - 7483 - 84 88 .094 299 -39464 -15371 44 .695 990 - 7567 - 92 89 .054 835 -54835 224 Bulletin of the Bureatt of Standards [ Vol. 3, No. I
TABLE XXI
Nagaoka's Table of Values of the End Correction K as Function of the Ratio Diameter Length For use in Formula (75))
Diameter Diameter
Length K Ai A2 Length K A! A 2
0.00 1.000 000 -4231 +24 0.45 0.833 723 -3160 +21
.01 .995 769 -4207 26 .46 .830 563 -3139 22
.02 .991 562 -4181 24 .47 .827 424 -3117 21
.03 .987 381 -4157 25 .48 .824 307 -3096 21
.04 .983 224 -4132 25 .49 .821 211 -3075 21
0.05 0.979 092 -4107 +25 0.50 0.818 136 -3054 +21
.06 .974 985 -4082 26 .51 .815 082 -3033 21
-07 .970 903 -4056 24 .52 .812 049 -3012 21
.08 .966 847 -4032 24 .53 .809 037 -2991 20
.09 .962 815 -4008 26 .54 .806 046 -2971 21
0.10 0.958 807 -3982 +25 0.55 0.803 075 -2950 +20
.11 .954 825 -3957 24 .56 .800 125 -2930 20
.12 .950 868 -3933 23 .57 .797 195 -2910 20
.13 .946 935 -3910 26 .58 .794 285 -2890 20
.14 .943 025 -3884 27 .59 .791 395 -2870 20
0.15 0.939 141 -3857 +23 0.60 0.788 525 -2850 + 19
.16 .935 284 -3834 23 .61 .785 675 -2831 19
.17 .931 450 -3811 26 .62 .782 844 -2812 20
.18 .927 639 -3785 24 .63 .780 032 -2792 19
.19 .923 854 -3761 24 .64 .777 240 -2773 19
0.20 0.920 093 -3737 +24 0.65 0.774 467 -2754 + 19
.21 .916 356 -3713 24 .66 .771 713 -2735 19
.22 .912 643 -3689 25 .67 .768 978 -2716 19
.23 .908 954 -3664 23 .68 .766 262 -2697 18
.24 .905 290 -3641 25 .69 .763 565 -2679 18
0.25 0.901 649 -3616 +23 0.70 0.760 886 -2661 + 18
.26 .898 033 -3593 24 .71 .758 225 -2643 19
.27 .894 440 -3569 23 .72 .755 582 -2624 17
.28 .890 871 -3546 24 .73 .752 958 -2607 18
.29 .887 325 -3522 24 .74 .750 351 -2589 18
0.30 0.883 803 -3498 +22 0.75 0.747 762 -2571 + 17
.31 .880 305 -3476 24 .76 ,745 191 -2554 17
.32 .876 829 -3452 23 .77 .742 637 -2537 18
.33 .873 377 -3429 23 .78 .740 100 -2519 17
.34 .869 948 -3406 22 .79 .737 581 -2502 16
0.35 0.866 542 -3384 +24 0.80 0.735 079 -2486 + 19
.36 .863 158 -3360 22 .81 .732 593 -2467 16
.37 .859 799 -3338 23 .82 .730 126 -2451 16
.38 .856 461 -3315 22 .83 .727 675 -2435 16
.39 .853 146 -3293 23 .84 .725 240 -2419 17
0.40 0.849 853 -3270 +22 0.85 0.722 821 -2402 + 16
.41 .846 583 -3248 23 .86 .720 419 -2386 16
.42 .843 335 -3225 21 .87 .718 033 -2370 15
.43 .840 110 -3204 21 .88 .715 663 -2355 16
.44 .836 906 -3183 23 .89 .713 308 -2339 17
Graver] Formulas for Mutual and Self-Inductance 225
TABLE XXI— Continued
Diameter Diameter Length K Aj A 2 Length K A, A 2 A 3
0.90 0.710 969 -2322 + 14 2.50 0.471 865 -9292 + 405 .91 .708 647 -2308 16 2.60 .462 573 -8887 378 .92 .706 339 -2292 15 2.70 .453 686 -8509 355 .93 .704 047 -2277 16 2.80 .445 177 -8154 330 .94 .701 770 -2261 14 2.90 .437 023 -7824 312
0.95 0.699 509 -2247 + 15 3.00 0.429 199 -7512 + 293 .96 .697 262 -2232 15 3.10 .421 687 -7219 275 .97 .695 030 -2217 15 3.20 .414 468 -6944 260 .98 .692 813 -2202 14 3.30 .407 524 -6684 245 .99 .690 611 -2188 14 3.40 .400 840 -6439 230
1.00 0.688 423 -10726 + 344 3.50 0.394 401 -6209 + 220 1.05 .677 697 -10382 330 3.60 .388 192 -5989 207 1.10 .667 315 -10052 316 3.70 .382 203 -5782 195 1.15 .657 263 -9736 303 3.80 .376 421 -5587 186 1.20 .647 527 -9433 290 3.90 .370 834 -5401 174
1.25 0.638 094 -9143 + 278 4.00 0.365 433 -5227 + 168 1.30 .628 951 -8865 266 4.10 .360 206 -5059 161 1.35 .620 086 -8599 255 4.20 .355 147 -4898 152
1.40 .611 487 -8343 244 4.30 .350 249 -4746 141
1.45 .603 144 -8099 236 4.40 .345 503 -4605 138
1.50 0.595 045 -7863 + 224 4.50 0.340 898 -4467 + 134
1.55 .587 182 -7639 215 4.60 .336 431 -4333 125 1.60 .579 543 -7424 208 4.70 .332 098 -4208 118
1.65 .572 119 -7216 198 4.80 .327 890 -4090 115
1.70 .564 903 -7018 190 4.90 .323 800 -3975 102
1.75 0.557 885 -6828 + 184 5.00 0.319 825 -18321 +2227 -397 1.80 .551 057 -6644 176 5.50 .301 504 -16094 1830 -306 1.85 .544 413 -6468 170 6.00 .285 410 -14264 1524 -241 1.90 .537 945 -6298 161 6.50 .271 146 -12740 1283 -193 1.95 .531 647 -6137 154 7.00 .258 406 -11457 1090 -153
2.00 0.525 510 -11809 + 580 7.50 0.246 949 -10367 + 937 -127 2.10 .513 701 -11229 539 8.00 .236 582 -9430 810- -104 2.20 .502 472 -10690 499 8.50 .227 152 -8620 706 - 86 2.30 .491 782 -10191 465 9.00 .218 532 -7914 620
2.40 .481 591 -9726 434 9.50 .210 618 -7294
10.00 0.203 324
In the last part of this table several errors in the fifth and sixth places of decimals have been corrected. 226 Bulletin of the Bureau of Standards {Vol. 8, No. i
TABLE XXII
Functions for Calculating Resistance and Inductance of Straight, Cylindrical Wires with Varying Frequency {sec. 10)
X W A A x W A, A Z A, A 4 Z A, A Y x 2 2 Y 2 Y 2 x Y 2
0.0 00 1.00000 + 1 0. +2500 1.00000 .1 20.00000 1.00000 + 1+2 0.02500 2500 1.00000 - 2 .2 10.00020 ... 1.00001 3+6 0.05000 2500 - 1 1.00000 - 2 - 3 .3 6.66693 1.00004 9+10 0.07500 2499 0.99998 - 5 - 5 .4 5.00065 1.00013 + 16 19 0.09999 2499 - 2 0.99993 - 10 - 8 0.5 4.00128 .... 1.00032 + 35+22 0.12498 +2497 - 3 0.99984 - 18 -11 .6 3.33557 1.00067 57 + 31 0.14995 2494 - 4 0.99966 - 29 -14 .7 2.86069 1.00124 88 + 40 0.17489 2490 - 7 0.99937 - 43 -20 .8 2.50530 — 1.00212 128 + 51 0.19979 2483 - 10 0.99894 - 64 -25 .9 2.22978 1.00340 179 + 60 0.22462 2473 - 12 0.99830 - 89 -31 1.0 2.01038 1.00519 + 239 + 74 0.24935 +2461 - 18 0.99741 - 120 -36 1.1 1.83196 1.00758 313 + 86 0.27396 2443 - 21 0.99621 - 156 -43 1.2 1.68451 1-01071 399 100 0.29839 2422 - 27 0.99465 - 199 -50 1.3 1.56108 1.01470 499 114 0.32261 2395 - 34 0.99266 - 249 -57 1.4 1.45670 1.01969 613 128 0.34656 2361 - 41 0.99017 - 306 -63 1.5 1.36776 1.02582 + 741 141 0.37017 +2320 - 48 0.98711 - 369 -69 1.6 1.29154 1.03323 882 153 0.39337 2272 - 56 0.98342 - 438 -76 1.7 1.22594 1.04205 1035 165 0.41609 2216 - 64 0.97904 - 514 -81 1.8 1.16934 1.05240 1200 176 0.43825 2152 - 72 0.97390 - 595 -86 1.9 1.12042 1.06440 1376 183 0.45977 2080 - 81 0.96795 - 681 -89
2.0 1.07816 -3649 +505 1.07816 1559 192 0.48057 + 1999 - 90 0.96113 - 770 -92 2.1 1.04167 -3144 442 1.09375 1751 192 0.50056 1909 - 96 0.95343 - 862 -92 2.2 1.01023 2702 387 1.11126 1943 195 0.51965 1813 -102 0.94482 - 954 -91 2.3 0.98321 2315 339 1.13069 2138 192 0.53778 1711 -108 0.93527 -1045 -90 2.4 0.96006 1976 297 1.15207 2330 188 0.55489 1603 -113 0.92482 -1135 -86 2.5 0.94030 -1679 +256 1.17538 2518 179 0.57092 + 1490 -115 0.91347 -1221 -81 2.6 0.92351 1423 224 1.20056 2697 170 0.58582 1375 -116 0.90126 -1301 -73 2.7 0.90928 1199 190 1.22753 2867 157 0.59957 1259 -116 0.88825 -1374 -65 2.8 0.89729 1009 162 1.25620 3024 142 0.61216 1143 -114 0.87451 -1439 -56 2.9 0.88720 847 136 1.28644 3166 128 0.62359 1029 -111 0.86012 -1495 -47 3.0 0.87873 - 711 +114 1.31809 +3293 +109 0.63388 + 918 -106 0.84517 -1542 -36 3.1 0.87162 597 92 1.35102 + 93 3402 0.64306 812 -100 0.82975 -1578 -25 3.2 0.86565 505 75 1.38504 3495 + 76 0.65118 712 - 93 0.81397 -1603 -16 3.3 0.86060 430 58 1.41999 3571 + 61 0.65830 619 - 87 0.79794 -1619 - 6 3.4 0.85630 372 46 1.45570 3632 + 44 0.66449 532 - 78 0.78175 -1625 + 4 3.5 0.85258 - 326 + 36 1.49202 +3677 + 31 0.66981 454 - 70 0.76550 -1621 +11 3.6 0.84932 290 24 1.52879 3708 + 19 0.67436 384 - 61 0.74929 -1610 +20 3.7 0.84642 266 18 1.56587 3727 + 10 0.67820 323 - 55 0.73320 -1590 26 3.8 0.84376 248 13 1.60314 3737 - 1 0.68143 268 - 47 0.71729 -1564 31 3.9 0.84128 235 8 1.64051 3736 - 7 0.68411 221 - 40 0.70165 -1533 36 GroLr\ Formulas for Mutual and Self-Inductance 227
TABLE XXII— Continued
W x W Z 4 Z
X Y Ai A 2 2 Y A x A 2 Y \ A 2 x Y Aj A 2
4.0 0.83893 - 227 + 5 1.67787 +3729 - 12 0.68632 + 181 - 33 0.68632 -1497 +39 4.1 0.83666 222 3 1.71516 3717 - 17 0.68813 148 - 28 0.67135 -1458 43 4.2 0.83444 219 + 1 1.75233 3700 - 19 0.68961 120 - 23 0.65677 -1415 43 4.3 0.83225 218 1.78933 3681 - 20 0.69082 97 - 17 0.64262 -1372 45 4.4 0.83007 218 1.82614 3661 - 22 0.69179 80 - 15 0.62890 -1327 45
4.5 0.82789 - 218 + 1 1.86275 +3639 - 20 0.69259 + 65 - 12 0.61563 -1282 +45 4.6 0.82571 217 + 1 1.89914 3619 - 21 0.69324 53 - 8 0.60281 -1237 45 4.7 0.82354 216 + 1 1.93533 3598 - 19 0.69377 45 - 6 0.59044 -1192 43 4.8 0.82138 215 + 1 1.97131 3579 - 17 0.69422 39 - 4 0.57852 -1149 43 4.9 0.81923 - 214 + 2 2.00710 +3562 - 15 0.69461 35 - 3 0.56703 -1106 42
5.0 0.81709 -419 +14 2.04272 + 7081 -45 0.69496 +62 -6 0.55597 -2091 +151 5.2 0.81290 405 18 2.11353 7036 -31 0.69558 56 -1 0.53506 1940 138 5.4 0.80885 387 20 2.18389 7005 -19 0.69614 55 0.51566 1802 124 5.6 0.80498 367 22 2.25393 6987 - 8 0.69669 55 +1 0.49764 1678 112 5.8 0.80131 345 23 2.32380 6979 0.69725 56 -1 0.48086 1566 101
6.0 0.79786 -322 +21 2.39359 + 6979 + 4 0.69781 +55 -1 0.46521 -1465 + 91 6.2 0.79464 301 21 2.46338 6983 8 0.69836 54 -2 0.45056 1374 82 6.4 0.79163 280 19 2.53321 6992 8 0.69891 52 -3 0.43682 1292 74 6.6 0.78883 261 17 2.60313 6999 8 0.69942 49 -3 0.42389 1218 68 6.8 0.78621 244 16 2.67312 7007 8 0.69991 46 -4 0.41171 1150 62
7.0 0.78377 -228 +13 2.74319 + 7015 + 6 0.70037 +42 -3 0.40021 -1088 + 57 7.2 0.78149 215 13 2.81334 7021 5 0.70080 39 4 0.38933 1031 52 7.4 0.77934 202 12 2.88355 7026 5 0.70118 35 3 0.37902 979 49 7.6 0.77731 190 11 2.95380 7031 3 0.70154 32 3 0.36923 930 45 7.8 0.77541 179 9 3.02411 7034 + 1 0.70185 29 -2 0.35992 885 42
8.0 0.77361 -170 + 8 3.09445 + 7035 + 3 0.70214 +27 -2 0.35107 - 843 + 39 8.2 0.77191 162 8 3.16480 7038 + 1 0.70241 25 -2 0.34263 804 36 8.4 0.77028 154 7 3.23518 7039 + 1 0.70265 23 -2 0.33460 768 34 8.6 0.76874 147 7 3.30557 7040 + 1 0.70288 20 -1 0.32692 734 32 8.8 0.76727 140 6 3.37597 7041 + 1 0.70308 19 -1 0.31958 702 30
9.0 0.76586 -134 + 6 3.44638 + 7042 + 1 0.70327 +18 -2 0.31257 - 672 + 28 9.2 0.76452 128 5 3.51680 7043 - 1 0.70345 16 1 0.30585 644 26 9.4 0.76324 123 6 3.58723 7043 + 2 0.70362 15 1 0.29941 617 25 9.6 0.76201 117 4 3.65766 7045 + 4 0.70377 14 1 0.29324 593 23 9.8 0.76084 -113 +4 3.72812 7046 + 2 0.70391 +13 '
-1 0.28731 - 569 + 21
10.0 0.75971 -261 +24 3.79857 +17620 + 3 0.70405 +30 -4 0.28162 -1330 +120 10.5 0.75710 237 21 3.97477 17623 4 0.70435 26 4 0.26832 1210 104 11.0 0.75473 216 19 4.15100 17627 4 0.70461 22 3 0.25622 1106 91 11.5 0.75257 197 16 4.32727 17631 4 0.70483 19 2 0.24516 1015 80 12.0 0.75060 181 15 4.50358 17635 3 0.70503 17 -2 0.23501 935 71
12.5 0.74879 -166 +13 4.67993 +17638 + 3 0.70520 +15 -1 0.22567 - 863 + 64 13.0 0.74712 154 11 4.85631 17641 2 0.70535 14 2 0.21703 800 57 13.5 0.74559 142 10 5.03272 17643 2 0.70549 12 1 0.20903 743 51 14.0 0.74416 132 9 5.20915 17645 3 0.70561 11 2 0.20160 692 46 14.5 0.74284 -123 + 8 5.38560 17648 + 2 0.70572 + 9 -1 0.19468 -646+40 .
228 Bulletin of the Bureau of Standards [ I ~ol. 8, No. i
TABLE XXII— Continued
X w Aj x W Z 4 Z A, Y ^2 2 Y Y Al "' i Y Ai
15.0 0.74161 -222 +27 5.56208 +35301 + 6 0.70581 +16 -3 0.18822 -1172 +137 16.0 0.73939 195 22 5.91509 35307 5 0.70597 13 2 0.17649 1035 114 17.0 0.73743 173 19 6.26817 35312 5 0.70611 11 1 0.16614 921 97 18.0 0.73570 154 16 6.62129 35317 4 0.70622 10 2 0.15594 824 82 19.0 0.73416 139 13 6.97446 35321 3 0.70632 8 -1 0.14870 742 70
20.0 0.73277 -125 +11 7.32767 +35324 - 3 0.70640 7 -1 0.14128 - 672 + 61 21.0 0.73151 114 10 7.68091 35327 2 0.70646 6 1 b. 13456 611 53 22.0 0.73038 104 9 8.03418 35329 2 0.70652 5 0.12846 558 46 23.0 0.72935 95 8 8.38748 35331 2 0.70657 5 1 0.12288 511 41
24.0 0.72840 87 7 8.74079 35333 2 0.70662 4 -1 0.11777 470 36
25.0 0.72753 - 80 + 5 9.09412 +35335 + 2 0.70666 +3 0.11307 - 434 + 32 26.0 0.72673 -143 +20 9.44748 +70674 + 5 0.70669 6 -1 0.10872 - 775 +103 28.0 0.72530 123 15 10.15422 70679 - 4 0.70675 5 —1 0.10096 672 84 30.0 0.72407 -108 -13 10.86101 +70683 - 3 0.70680 +4 -1 0.09424 - 589 + 69 32.0 0.72299 95 11 11.56785 70686 3 0.70684 3 0.08835 519 58
34.0 0.72204 84 9 12.27471 70689 2 0.70687 2 0.08316 462 49
36.0 0.72120 75 8 12.98160 70691 2 0.70689 2 0.07854 413 41
38.0 0.72045 68 7 13.68852 70693 2 0.70691 2 0.07441 372 35
40.0 0.71977 -61 + 6 14.39545 + 70695 1 0.70693 +2 0.07069 - 336 + 30 42.0 0.71916 55 5 15.10240 70696 2 0.70695 1 0.06733 306 27
44.0 0.71861 50 4 15.80936 70698 1 0.70696 2 0.06427 279 23 46.0 0.71810 46 4 16.51634 70699 0.70698 1 0.06148 256 20 48.0 0.71764 - 43 + 3 17.22333 + 70699 + 1 0.70699 +1 0.05892 - 236 + 17 50.0 0.71721 -170 +49 17.93032 +353509 +13 0.70700 +3 0.05656 - 942 +269 60.0 0.71551 121 30 21.46541 353522 8 0.70703 2 0.04713 673 168
70.0 0.71430 91 20 25.00063 353530 5 0.70705 1 0.04040 505 112
80.0 0.71340 70 + 14 28.53593 353535 + 3 0.70706 1 0.03535 393 79
90.0 0.71270 - 56 32.07127 353538 . 0.70707 +1 0.03142 - 314 + 55 100.0 0.71213 35.60666 0.70708 0.02828 oo 0.70711 oo 0.70711 0. GrTver] Formulas for Mutual aud Self-Inductance 229
TABLE XXIII
Values of Limiting Change of Inductance with the Frequency
Single Wire Parallel Wires Circular Rings
21 d 8a
P
A2
P /MA P /AL
A
\L /x=oo Ai Ai A2 A2
VL /x=oo VL /x=oo t
50 0.07906 5 0.13445 -1201 + 342
6 .12244 859 206
100 0.06485 -988 +538 7 .11385 653 137 100 0.08756 -1710 +987
200 5497 450 173 8 .10732 516 84 200 7046 723 294
300 5047 277 82 9 .10216 - 432 300 6323 429 134
400 4770 195 47 400 5894 295 75
500 4575 -148 + 30 10 0.09784 -2078 +1219 500 5600 - 220 + 47
20 7706 859 359
600 0.04427 -118 + 21 30 6847 500 160 600 0.05380 - 173 + 32
700 4309 97 15 40 6347 340 88 700 5207 141 23
800 4213 82 11 50 6007 - 252 + 55 800 5066 118 17
900 4131 -71+9 900 4948 - 101 + 13
60 0.05755 - 197 + 37
1000 0.04060 -411 +207 70 5557 160 26 1000 0.04847 - 574 +297 2000 3649 204 72 80 5397 134 20 2000 4273 277 101 3000 3445 131 36 90 5263 - 114 + 15 3000 3996 176 50 4000 3314 95 22 4000 3820 126 29 5000 3219 -74+14 100 0.05149 - 643 + 336 5000 3694 - 97+19 200 4506 307 113 6000 0.03145 -60+10 300 4199 194 56 6000 0.03597 - 78+13 7000 3085 50 7 400 4005 138 32 7000 3519 65 10 8000 3035 43 6 500 3867 - 106 + 21 8000 3454 55 8 9000 2992 -37+4 9000 3399 -48+6 600 0.03761 - 85 + 14 10000 0.02955 -224 +108 700 3676 71 11 10000 0.03351 - 285 +140 20000 2731 116 40 800 3605 60 8 20000 3066 145 50 30000 2615 76 20 900 3545 - 52 + 6 30000 2921 95 25 40000 2539 56 12 40000 2826 70 16 50000 2483 -44+8 1000 0.03493 - 309 + 154 50000 2756 - 54+10 2000 3183 155 53 60000 0.02438 -36+6 3000 3028 102 27 60000 0.02702 -44+7 70000 2402 30 4 4000 2926 75 17 70000 2658 37 ' 5
80000 2372 26 3 5000 2851 - 58 + 11 80000 2621 32 4 90000 2346 -23+2 90000 2589 -28+3 6000 0.02793 - 47 + 7 100000 0.02323 7000 2746 40 6 100000 0.02561 1000000 1913 8000 2706 34 4 1000000 2072 9000 2672 - 30 + 3 10000 2643 230 Bulletin of the Bureau of Standards [ Vol. 8, No. /]
TABLE XXIV
Values of the Argument x for Copper Wires 1 mm Radius and Conductivity 5.811x10' c g. s. Units
t xo A f x« K
cycles per meters cycles per meters second second
100 0.2142 50000 4.790 6000
200 .3029 60000 5.247 5000
300 .3710 70000 5.667 4286
400 .4284 80000 6.058 3750
500 .4790 90000 6.426 3333
600 0.5247 100000 6.774 3000
700 .5667 150000 8.296 2000
800 .6058 200000 9.579 1500
900 .6426 250000 10.710 1200
1000 .6774 300000 11.732 1000
2000 0.9579 333333 12.367 900
3000 1.1732 375000 13.117 800
4000 1.3547 428570 14.023 700
5000 1.5146 500000 15.146 600
600000 16.592 500
6000 1.6592 700000 17.921 429
7000 1.7921 750000 18.550 400
8000 1.9158 800000 19.158 375
9000 2.0321 900000 20.321 333
1000000 21.42 300
10000 2.142 30000 1500000 26.23 200
15000 2.623 20000 3000000 37.10 100
20000 3.029 15000
30000 3.710 10000
40000 4.284 7500
INDEX [Italicized page numbers refer to examples illustrating the formulas. Proper names are also itali- cized.]
Absolute formulas Mutual inductance of coaxial circles, 6, 7, 9, 20, 21\ 23; mu- :
tual inductance of concentric, coaxial solenoids, 64, 73, 78, gi; mutual inductance of coaxial solenoids, 64, 69, 73, 89; self-inductance of solenoids, 117, 118, 129, 132, 733, Table IV; mutual inductance of circle and solenoid, 99, 100, 103, 706, io/y 708, 709; inductance of rectangle of rectangular section, 155. Absolute invariant 17. Adjacent conductors. See Linear conductors. Airey. See Searle and Airey. Ampere balance. See Current balance. Amplitude of incomplete elliptic integrals, 64, 100. Annular area, geometric mean distance of, 168, 169; geometric mean distance of point to area, 169. Annulus. See Annular area. Approximate formulas; inductance of solenoid of more than one layer, Cohen, 140, 750; inductance of circular ring, Kirchhoff, no, 774; mutual inductance of coaxial solenoids of equal length, 55, 78; inductance of coil of rectangular cross section, Maxwell, 135, 743; Perry 126, 743; mutual inductance of coaxial circles, Wiede- mann, footnote 13. Arithmetical mean distance, 171, 172. Arithmetical mean square distance, 171, 172. Asymptotic formulas for IV, Y, and Z, 176. Attraction of coils. See Current balance. Ayrton and Jones current balance, 704.
B
Bar. See Rectangular bar. Ber and bei functions, 174, 175. Bessel functions, 15, 174. Blathy, inductance of a ring, 113. Breadth, equivalent of coil 38, 47, Bromwich, 16, in, 113.
Campbell, extension of Joneses formula, 100, 709; form for standard of mutual in- ductance, 708. Choice of formulas: Mutual inductance of coaxial circles, 19; mutual inductance of coaxial coils of rectangular section, 43; mutual inductance of coaxial solenoids, 73, 83, 84, 86; inductance of solenoids, 125; inductance of coils of rectangular cross section, 142. Circles. See Coaxial circles,, Circular areas, Circle and coaxial solenoid. Circle and coaxial solenoid, mutual inductance of, 98-110. Circular areas: Geometric mean distance of, 167; geometric mean distance of a point from, 169; geometric mean distance of two, 170; arithmetical mean distance of, 171; arithmetical mean square distance of, 171. 231 232 Index
Circular coils of rectangular cross section Inductance of, 135-150 choice of formu- : ;
las, 142 mutual inductance of coaxial, 33-52 choice of formulas, 42. ; ;
Coaxial circles Formulas for the inductance of, 6-32 choice of formulas, 19 sum- : ; ;
mary of formulas, 19. Coaxial coils of rectangular cross section Formulas for the mutual inductance of, :
33-52 choice of formulas, 42. ;
Coaxial solenoids Formulas for the mutual inductance of, 52-98 case where coils : ;
are not concentric, 59, 64, 73 choice of formulas, 73, 83, 84, 86. ;
Coefficients: In Stefan's formula (90), 196; in hypergeometrical series of Mathy formula (18), 220; in formulas (40), (43), and (56), 221, 222. Coffin, Absolute formula for mutual inductance of equal circles, 14, 15, 30; coaxial solenoids, 73 inductance of solenoid, 117, 129 derivation of Lorenz" s formula, 118. ; ;
Cohen, absolute formula for the mutual inductance of coaxial solenoids 64, 69, 73, 78, 79, 91 ; correction of JVien's formula, in approximate formula for the induc- ;
tance of solenoids of several layers, 140, 150. Complementary modulus, 8, 10, 11. Complete elliptic integrals. See Elliptic integrals. Concentric coaxial solenoids. See Coaxial solenoids. Concentric conductors, inductance of, 158 with high frequency, 179. ;
Conductors. See Concentric conductors, linear conductors, rings, tubes, tapes, etc. Constant of Lorenz apparatus. See Lorenz apparatus. Constants. See Tables of constants, etc. Correction, of current sheet formulas for inductance of solenoids for windiug of round wire, 122, 127 ; for unequal distribution of current over cross section of coil, 140-142, 147-149; of simple expression for toroidal coil, 125. Correction factor. See End correction, correction, etc. Cross section. See Coaxial coils of rectangular cross section, Circular coils of rec- tangular cross section, Equal coils of rectangular cross section, Square cross section. Current balance, of Ayrton and. Jones, 104, 106; of National Physical Laboratory, 107. Current sheets, 76, 97, 119. Cylindrical wire. See Straight cylindrical wire. Cylindrical coils. See Solenoids, Coaxial solenoids. Cylindrical conductors. See Linear conductors.
Decrease of inductance with the frequency General considerations, 172 of straight : ;
cylindrical wires, 173, 174, 177, 179, 183-185 ; of two parallel cylindrical wires, 180, 181, 185, 186 ; of a circular ring, 181, 182, 186, 187. Differential coefficients, 34, 39. Dimensions of equivalent current sheet 76, 97, 119. Disk in Lorenz apparatus. See Lorenz apparatus. Distribution of current. See Correction for unequal, High frequency formulas, In- crease of resistance, Decrease of inductance. Dynamometers: Gray, 60, 86; Ayrton and Jones, 104, 106.
E
"Eddy currents. See High frequency formulas, Decrease of inductance, Increase of resistance. Ellipse. See Elliptical area. Elliptical area, geometric mean distance of, 167. Elliptic integrals. See also Incomplete elliptic integrals, mutual inductance formu- las involving 6, 7, 9, 64, 68, 71, 98, 99, 100; inductance of solenoids, 118; series expansions for complete, 8, 9; tables of, as functions of tan y, 193; Legendre's Tables, 202-212. End correction For mutual inductance of equal concentric solenoids, 55 for self- : ;
inductance of solenoids, 1 19-12 1, 130-132, 223-225. End effect. See End correction. Index 233 Equal circles, mutual inductance of 14, 28, 30, 18. Equal coils of rectangular cross section, mutual inductance of, 33, 39, 40, 44, 45, 47, 49-51- Equal parallel rectangles. See Parallel rectangles. Equal squares. See Squares, Geometric mean distance. Equal solenoids, mutual inductance of, 69-71, 94-97. Equal radii. See Equal circles, Equal coils of rectangular cross section, Equal solenoids, Solenoids of equal radii. Equal cross section. See Equal coils of rectangular cross section. Equivalent circles. See Equivalent filaments. Equivalent breadth. See Equivalent filaments. Equivalent radius. See Equivalent filaments. Equivalent length. See Dimensions of equivalent current sheet. Equivalent filaments, 38, 39, 47, 48. Errors in Rowland's and Rayleigh''s formulas, 36, 37. Exact formulas. See Absolute formulas. Examples illustrating the formulas For mutual inductance of coaxial circles, 20-32; :
mutual inductance of coaxial coils of rectangular cross section, 44-52; mutual in- ductance of coaxial solenoids, 77-98 ; mutual inductance of circle and solenoid, 103-110; self -inductance of circular ring, 774-115 ; self-inductance of solenoids, 126-135 ; self-inductance of coils of rectangular cross section, 142-150; self and mutual inductance of linear conductors, 159-166 ; inductance and resistance with high frequency, 183-187. Extension of Maxwell's series formula for circles, 14,30; BlaxwelVs formula for equal, concentric solenoids, 53, 77-80 ; Rbiti's formula for coaxial solenoids, 57-59, 80-86 ; Jones's formula for circle and solenoid, 10 1, 102, 103-106, 108, 109 ; Ray- leigh' s and Nivens's formula for the inductance of solenoids, 117, 129 ; Russell's formulas for W, Y, and Z, 176.
Filaments. See Equivalent filaments. Force of attraction of coils. See Dynamometers, Current balances. Formulas. See Absolute formulas, Approximate formulas, Correction formulas, In- terpolation formulas. Frequency. See High frequency formulas. Frohlich, inductance of toroidal coil, 125. Functions. See Bessel functions, p function, q series, Ber and bei functions, Ker and kei functions, tables, coefficients, W. Y. and Z.
General term: In Wallis's formulas for Fand F, 9; in formulas (5) and (6), 9, 10; in Havelock's formula for coaxial solenoids, 56; in Gray's and Searle and Airey's formulas, 63; in Russell's formula for coaxial solenoid, 68; in Dorenz's formula for circle and solenoid, 99; in the Webster-Havelock formula, 121; in Russell's formulas for IF, Y, and Z, 176. Geometric mean distance, 42, 52, 166-170. Glazebrook, 34. Gray, mutual inductance of coaxial solenoids, 59, 60, 86-89; dynamometer, 60, 86; geometric mean distance formulas, 170.
Harmonics. See Zonal harmonics. Havelock, mutual inductance of coaxial circles, 15, 16, 27 29, 30; mutual inductance ',
of coaxial solenoids, 55, 56, 72, 78; mutual inductance of short secondary on long primary, 68; self-inductance of solenoids (see Webster). Heaviside, equal coaxial solenoids, 55, 78; high frequency formulas, 173. Hicks, inductance of a ring, 113. High frequency formulas, in, 172-187. 234 Index
Himstedt, mutual inductance of solenoids, 72. Hollow tube. See Tube. Hypergeometric Series, 17, 56, Table XVII.
Illustration of formulas. See Examples for illustrating the formulas. Incomplete elliptic integrals, 64, 67, 100. Increase of resistance with the frequency, 172-174, 177, 178-1S3, 183-/87, Table XXII. Infinite solenoid, inductance of, 116. Insulation of wires, correction for, 138, 139, 140-142. Integrals. See Elliptic integrals, Incomplete elliptic integrals. Interpolation: In Tables XV and XVI, 12; in Table XXII, 177; formula for, 214. Invariant. See Absolute invariant.
Jacobi, q series, 11, 12, 65-67, 120, 121; Theta functions, 66. Jones, mutual inductance of circle and solenoid, 99, 102, 106; see also Ayrton, Campbell. Joubert. See Mascart.
YLelvin, resistance and inductance of straight wires at high frequency, 173, 174; ber and bei functions, 175. Ker and Kei functions, 175. Kirchhoff, inductance of a ring, no, 114; inductance of a solenoid, 118; summation formula for the inductance of a solenoid, 123; inductance of a square, 154; formula for mutual inductance of coaxial coils, 73.
L
,anden s transformation, 7. y
Layers, coils of several. See Solenoids. Legendre 's tables, 6, 8, 10, 20, 44, 100, Tables XII and XIII. Limiting formulas for resistance and inductance of straight wires with the fre- quency, 177. Line, geometric mean distance of, 167; geometrical mean distance of two lines, 168, 169, 170; arithmetical mean distance of, 171; arithmetical mean square distance of, 171. Linear conductors, self and mutual inductance of, 150-159, 139-166; mutual induc- tance of two conductors in the same straight line, 152, 161. Logarithms. See Natural logarithms. Long coils, mutual inductance, choice of formulas, 73-77; self -inductance of, 116, 117, 120, 129, 130-133. Loretiz, apparatus, 34, 103, 107; mutual inductance of circle and solenoid, 9S; induc- tance of solenoids, 117, 118, 129, 130, 132, 133. Lyle, mutual inductance of coils of rectangular cross section, 38, 47, 48.
M
Marlens, mutual inductance of rectangles, 156. Mascart, 55. Jlathy, corrected formula of, for coaxial circles, 17, 31; simple special formula de- rived from this, 18, 32. Maxwell, mutual inductance of coaxial circles, 6-8, 13, 20, 21, 28, Table I; mutual inductance of equal coaxial, concentric solenoids, 53, 77, 79, 80; inductance of a ring, no, in, 114, 113; inductance of coil of rectangular cross section, 135, 136, j 42, j 43; correction for distribution of current in round wires, 140. Index 235
McGill University, Lorenz apparatus of, 103. Method of obtaining dimension of coils. See Dimensions of equivalent current sheets. Minchin, inductance of ring, 113. •
Misprints in authorities quoted, 34, 55, 64, 100, 103, 170. Modified Radius. See Equivalent filaments. Modulus. See Elliptic integrals, Incomplete integrals, Complementary modulus. Multiple conductors, inductance of, 159, 165. Mutual inductance. See detailed headings, such as Coaxial circles, Coaxial sole- noids, etc. Mutual inductance by means of self-inductance formulas, 71, 95, 96.
N
IXagaoka, mutual inductance of coaxial circles, 11, 12, 26, 2/, Table XV and XVI; mutual inductance of coaxial solenoids 64-67, 73, 89, 91-93; inductance of sole- noids, 119-121, 130-132. Nasmyth, 87. National Physical Laboratory, current balance of, 107. Natural logarithms of numbers from 1 to 100, Table XL Neumann, inductance of straight cylindrical wire, 151, 159; mutual inductance of parallel rectangles, 156, 164. Nicholson, inductance and resistance of parallel wires at high frequency, 180. Niven. See Rayleigh. Noninductive Shunts, inductance of, 158.
Ohm, determination of, 34 see also Lorenz apparatus. ;
Olshausen, absolute formulas for mutual inductance of coaxial coils, 73.
p function of Weierstrass, 17, 65. Parallel bars. See Rectangular bars. Parallel circles. See Coaxial circles. Parallel lines. See Lines. Parallel rectangles. See Rectangles. Parallel squares. See Squares. Parallel wires, mutual inductance of, 151, 160 ; see also Return circuit. Parameter, in Olshausen 's formula, 73. Permeability. See High frequency formulas, Linear conductors. Perry, inductance of coil of rectangular cross section, 136, 143. Pitch of winding, 76, 9/, 119. Plane, coaxial circles in the same, 14, 15, 18.
q Series of Jacobi, 11, 65-67. Quadratures, formula of, 34, 35, 45, 46, 90.
"Rayleigh (also Rayleigh and Niven), formula of quadratures, 34, 35, 45, 46, 90 ; in- ductance of circular ring, in, 114, 115 ; inductance of solenoids, 116, 126-129 ; in- ductance and resistance of straight wires at high frequency, 173, 177. Rectangles, geometric mean distance of, see Rectangular area inductance of, 154, ;
155, 163, 164; mutual inductance of parallel rectangles, 155, 164. Rectangular areas, geometric mean distance of, 167, 168 geometrical mean distance ;
of two, 170. 236 Index
Rectangular bars, inductance of, 152, 153, 162 ; mutual inductance of, 153, 154. Rectangular cross section. See Coaxial coils of rectangular cross section, Equal coils of rectangular cross section. Reduction from current sheet to winding of round wires, 122, 128^ Tables VII and VIII. Resistance. See Increase of resistance with the frequency. Return Circuit, inductance of parallel wire, 151, 152, 161 ; same at high frequency, 1S0, 181, 183, 186 ; inductance with rectangular cross section, 154, 162. Ring, inductance of circular solid, 110-112, 114; inductance with elliptical cross section, 113 inductance and resistance with high frequency, 181, 182, 186. ;
Rbiti, Mutual inductance of coaxial solenoids, 57, 58, 80-84, $7- Rosa, extension of Maxwell's formula for coaxial coils, 14, 30; mutual inductance of coils of rectangular cross section, 39, 49, §0 ; Rosa- Weinstein formula for coaxial coils, 40, 41, 31, 69-71, 94-97 ; extension of Searle and Airey 's formula 61-63, $2~ 86, 94; mutual inductance of circle and solenoid, 101, 102, 103, 105, 109 ; correc- tion for inductance of toroidal coil, 125, 134, 13s/ correction of current sheet for- mulas for winding of round wires, 122, 128-130, Tables VII and VIII correction ;
for distribution of current in coil of rectangular cross section, 138, 139, 141, 144- 150, Tables IX and X geometric mean distance, 168-170 arithmetical mean dis- ; ;
tance and arithmetical mean square distance, 171, 172. Rowland, mutual inductance of coaxial coils, 33, 34, 44, 43. Russell, mutual inductance of coaxial solenoids, 67, 68, 69, 83; self-inductance of solenoids, 121; formulas for the functions W, V, and Z, 175, 176; inductance and resistance of concentric main, 179.
Savidge, tables of functions for high frequency calculations, 175, 176. Searle and Airey, mutual inductance of coaxial solenoids, 61-63, 82-86, 94. Self-inductance, mutual inductance by means of, 41, 32, 71, 93, 96; see also detailed headings as Solenoids, Circular coils of rectangular cross sections, etc. Series formulas for F and E, 8, 9, 22, see Hypergeometric Series, W, Y and t Z functions, and other detailed headings. Short coils, inductance of short solenoids, 116, 120, 126, 128, 130. Short secondary on long primary, mutual inductance of, 68, 69, 93, 94. Shunts. See Noninductive shunts. Simple formula for certain coaxial circles, 18, 32. Single layer coils. See Solenoids, Circle and coaxial solenoids. Solenoids, inductance of infinite, 116; inductance of, 1 16-126, 126-133; inductance of solenoid of more than one layer 138-140, 130; see also Coaxial solenoids. Solid Ring. See Ring. Squares, inductance of, 154, 162, 164; mutual inductance of parallel, 155, 156, 164; geometric mean distance of, 167. Square cross section, mutual inductance of coaxial coils of, 39, 41, 4J, 49. Stefan, mutual inductance of coaxial coils, 40; inductance of circular coil of rectan- gular cross section, 137, 144-147, 149; correction for distribution of current in round wires, 141. Stirling's constants, 17, 19. Straight cylindrical wire, inductance of, 150, 151, 139, 160; inductance and resist- ance of, at high frequency, 173-179, 183-183, Tables XXII, XXIII. Strasser, inductance of single layer coil, 123, 124, 134, 14J, Table V. Strip. See Tape. Subdivision, calculation of inductance by, 43, 74, 88-90. Summation formula for inductance of solenoid, 123, 12J. Sumec, inductance of rectangle of rectangular cross section, 155.
Tables of Constants, 189-230. For detailed summary see Table of contents. Tape, winding of 119; self and mutual inductance of, 156, 157, 163; geometric mean distance of, 168, 169. Index 237
Taylor's theorem, 33. Terezawa, inductance of ring, footnote in. Theta functions, 66. Thick current sheet, 138, 139. Thin strip. See Tape. Thin-walled Tube. See Tube. Thomson, J. J., inductance of ring of elliptical section, 113. Toroidal coil, inductance of, 124, 125, 134. Tube, inductance of hollow, tubular ring, 112, 113, 115; straight tube, see High fre- quency formulas. u Unequal circles. See Coaxial circles. Unequal cross section, choice of formulas for mutual inductance of coils of, 43. Uniform magnetic field, inductance of ring rotating in, 113, 114. Uniform distribution of current in cross section, 140, 141, 147-149; see also High frequency formulas. Uniform winding. See Dimensions of equivalent current sheets.
w
W, T, and Z functions, definitions of, 174; expansions for, 176, 177; see also Table XXII. Wallis, series expansions for elliptic integrals, 9. Webster, inductance of a solenoid, 121, 132. Weierstrass, p functions, 17, 65. _ Weinstein, mutual inductance of coaxial circles, 7, 25 ; mutual inductance of coaxial coils, 40, 41,57, 71, 96; inductance of coil of rectangular cross section, 137, 143. Wiedemann, mutual inductance of circles, footnote, 13. Wien, inductance of ring, in, 112 ring rotating in uniform magnetic field, 113, 114. ;
Y
Y function. See W, Y, and Z functions. z Z function. See W, F, and Z functions. Zonal harmonics, 59. 21674 — 12 16