The self and mutual-inductance of linear conductors . WASHINGTON GOVERNMENT PRINTING OFFICE1908 THE SELF AND MUTUAL INDUCTANCES OF LINEAR CONDUCTORS. By Edward B. Rosa. Formulae for the self and. mutual inductances of straight wiresand rectangles are to be found in various books and papers, buttheir demonstrations are usually omitted and often the approximateformulae are given as though they were exact. I have thought thata discussion of these formulae, with the derivation of a number ofnew expressions, would be of interest, and that illustrations of theformulae, with some examples, would be o

The self and mutual-inductance of linear conductors . WASHINGTON GOVERNMENT PRINTING OFFICE1908 THE SELF AND MUTUAL INDUCTANCES OF LINEAR CONDUCTORS. By Edward B. Rosa. Formulae for the self and. mutual inductances of straight wiresand rectangles are to be found in various books and papers, buttheir demonstrations are usually omitted and often the approximateformulae are given as though they were exact. I have thought thata discussion of these formulae, with the derivation of a number ofnew expressions, would be of interest, and that illustrations of theformulae, with some examples, would be o Stock Photo

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The self and mutual-inductance of linear conductors . WASHINGTON GOVERNMENT PRINTING OFFICE1908 THE SELF AND MUTUAL INDUCTANCES OF LINEAR CONDUCTORS. By Edward B. Rosa. Formulae for the self and. mutual inductances of straight wiresand rectangles are to be found in various books and papers, buttheir demonstrations are usually omitted and often the approximateformulae are given as though they were exact. I have thought thata discussion of these formulae, with the derivation of a number ofnew expressions, would be of interest, and that illustrations of theformulae, with some examples, would be of service in making suchnumerical calculations as are often made in scientific and technicalwork. I have derived the formulae in the simplest possible manner, usingthe law of Biot and Savart in the differential form instead of Neu-manns equation, as it gives a better physical view of the variousproblems considered. This law has not, of course, been experi-mentally verified for unclosed circuits; but the self-inductance of anunclosed circuit means siselfmu430134419088080unse

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RM2AM5C9G–The self and mutual-inductance of linear conductors . use of the lawof Biot and Savart to obtain the self-inductance of an unclosed cir-cuit is perfectly legitimate. I have also shown how, by the use ofcertain arithmetical mean distances in addition to geometrical meandistances, the accuracy of some of the formulae can be increased. In the following demonstrations the magnetic field is assumed tobe instantaneous; in other words, the dimensions of the circuit areassumed to be small enough and the frequency of the current slow 302 Bulletin of the Bureau of Standards. [ Vol. 4, No. 2. enougli so

RM2AM5C1X–The self and mutual-inductance of linear conductors . ate the expression for the mag-netic force between the wires due to unit current, //= 2/^.Thus, id Ar=i I ?^=2/iog^ f Multiplying this by two (for both wires) and adding the term dueto the magnetic field within the wires we have the result given by(14). If the end effect is large, as when the wires are relatively farapart, use the expression for the self-inductance of a rectanglebelow (24); or, better, add to the value of (14) the self-inductance ofAB + CD using equation (10) in which /= 2AB. 4. MUTUAL INDUCTANCE OF TWO PARALLEL WIRES BY NE

RM2AM5BE8–The self and mutual-inductance of linear conductors . hence the actual self-inductance of such a cir-cuit is not the value of the self-inductance given byequation (9). The latter is the self-inductance of a part of a closed circuit due tothe current in itself. The actual self-inductance of any closed cir-cuit of which it is a part will be the sum of the self-inductances ofall the parts, plus the sum of the mutual inductances of each oneof the component parts on all the other parts. Thus the self-inductance of a rectangle is the sum of the self-inductances of thefour sides (by equation 10) plus

RM2AM5B6K–The self and mutual-inductance of linear conductors . hat case being 13.190) Z= 2(14.812—13.190) = 3.244 microhenrys. These calculations are of course all based on the assumption of auniform distribution of current through the cross section of the con-ductors. For alternating currents in which the current density isgreater near the surface the self-inductance is less but the mutualinductance is substantially unchanged. ^ Rosa, this Bulletin, 3, p. i. ^ Its more exact value is 2.0010, this Bulletin, 3, p. 9. Rosa. Inductance of Linear Conductors. Z^l 9. SELF-INDUCTANCE OF A SQUARE. The self-in

$The self and mutual-inductance of linear conductors . Fig. 25. where r^ is the g. m. d. of a single conductor ( = 0.7788/9, p being itsradius) and r^g is the distance between centers of the conductors iand 2, etc. If a is the radius of the circle on which the conductorsare distributed C. E. Guye, Comptes Rendus, 118, p. 1329; 1894. j^osa.] hiductance of Linear Conductors. 335 log 7?= log {r^na^^ or R = (r^ na-y ^^^) . The proof of this, as given by Guye, depends on the followingtheorem: If the circumference of a circle is divided into n equal parts bythe points A, B, C, . . and M be any point Stock Photo$

RM2AM59F7–The self and mutual-inductance of linear conductors . Fig. 25. where r^ is the g. m. d. of a single conductor ( = 0.7788/9, p being itsradius) and r^g is the distance between centers of the conductors iand 2, etc. If a is the radius of the circle on which the conductorsare distributed C. E. Guye, Comptes Rendus, 118, p. 1329; 1894. j^osa.] hiductance of Linear Conductors. 335 log 7?= log {r^na^^ or R = (r^ na-y ^^^) . The proof of this, as given by Guye, depends on the followingtheorem: If the circumference of a circle is divided into n equal parts bythe points A, B, C, . . and M be any point

RM2AM5BM2–The self and mutual-inductance of linear conductors . Bulletin of the Bi^reau of Standards. [ Vol. 4, No. 2. Fig. 4a. M=2. [/1o,^±4±^ V/m:Z+^] wliich is the same expression (12) foundby the other method. That process ismore direct and simpler to carry out thanto use Neumanns formula. 5. MUTUAL INDUCTANCE OF TWO LINEAR CON-DUCTORS IN THE SAME STRAIGHT LINE We have found the self-inductance of thefinite linear conductor AB by integratingthe magnetic force due to unit current inAB over the area ABBA, extending tothe right to infinity, equations (3) and (9). In the same way we may find themutual i

RM2AM59X3–The self and mutual-inductance of linear conductors . Fig. 22. Z=2/ hS+i] (56) This result can also be obtained by integrating the expression forthe force outside a^ between the limits a^ and a^^ and adding theterm (equation 6) for the field within a^^ there being no magneticfield outside a^. Thus U ax +i = ^^[ little greater, the geometrical mean distance from a^ to the tube,which is now more than a^ and less than a^, being given by theexpression <23 logg—«a log <^2 log a,j CL^ ^2 (57) Putting this value of log a in (56) in place oflog ^, we should have the self-inductance ofthe retu

$The self and mutual-inductance of linear conductors . TraS^ — I 2a cos 6. 2ad6 — /^a4.a .: S,= IT (50) Since the a. m. d. is the same for every pointof the circle we have also Fig. 19. s,= 4^ TT (51) For the arithmetical mean square distance we have nraS^ i 4^ 2 PB = a^--d^-{-2ad COS 6S, = a I {a--d-^2adcose)d0 = 7ra(d^-i-a^ TT^O, = ..S, = d^a (53). Fig. 20. For the entire area of the circle with respect to the point P •. S,^d+- (54) a If d= <9, Sj^ = —, the value for the area of the circle with respect to2 the center of the circle. For the area of a circle with respect toitself, the a. m Stock Photo$

RM2AM5AAG–The self and mutual-inductance of linear conductors . TraS^ — I 2a cos 6. 2ad6 — /^a4.a .: S,= IT (50) Since the a. m. d. is the same for every pointof the circle we have also Fig. 19. s,= 4^ TT (51) For the arithmetical mean square distance we have nraS^ i 4^ 2 PB = a^--d^-{-2ad COS 6S, = a I {a--d-^2adcose)d0 = 7ra(d^-i-a^ TT^O, = ..S, = d^a (53). Fig. 20. For the entire area of the circle with respect to the point P •. S,^d+- (54) a If d= <9, Sj^ = —, the value for the area of the circle with respect to2 the center of the circle. For the area of a circle with respect toitself, the a. m

$The self and mutual-inductance of linear conductors . log I (64) Fig. 26. w here since the g. m. d. of the central conductor oneach of the others is a. The self-inductance of the return system is A = 2/ logf-^] a. log 2/—log {f7ia ^) — I2/ log 2/ 1 a - I log --log (^^1^^^ ) + 1 4 (65) A=2/M=2l.. Z=2/ For a = 2 cm, a^—i cm, ^0 = 0.5, ;/ = 6, /= 1000 cm Z=: 2000 log, 4-loge (2.0525) + ^= 20oox 0.9173= 1.835 microhenrys. If the inner conductor were surrounded by a very thin tube ofradius 2 cm for a return, in place of the six wires, the self-inductanceof the return circuit would be [? Z=2ooo Stock Photo$

RM2AM5926–The self and mutual-inductance of linear conductors . log I (64) Fig. 26. w here since the g. m. d. of the central conductor oneach of the others is a. The self-inductance of the return system is A = 2/ logf-^] a. log 2/—log {f7ia ^) — I2/ log 2/ 1 a - I log --log (^^1^^^ ) + 1 4 (65) A=2/M=2l.. Z=2/ For a = 2 cm, a^—i cm, ^0 = 0.5, ;/ = 6, /= 1000 cm Z=: 2000 log, 4-loge (2.0525) + ^= 20oox 0.9173= 1.835 microhenrys. If the inner conductor were surrounded by a very thin tube ofradius 2 cm for a return, in place of the six wires, the self-inductanceof the return circuit would be [? Z=2ooo

$The self and mutual-inductance of linear conductors . -p LJ b log ^ |^H-|0 -4|^«log L ?b —yja^-^b-^b ] Putting ^a^--b^ — d^ the diagonal of the rectangle, 2ab . , 3<2 Z =: 4 <^ lop^- .+ 4 ^log- ^^ •^- 44 -^+/^] <f—^ —«+p or p{b-{-d)Z = 4 (<2-[-^)log a log (a-{-d) — b log (/^+(3^) 319 (24) For ^ = 200 cm, b= 100, /3 = o.i ^ = 8017.1 cm = 8.017 microhenrys.((^) 772^ conductor having a rectangular sectio7i. For a rectangle made up of a conductor of rectangular section, aX/S, A=2 r 1 2^ ; a log- ; ?^H-^+o.2235(a+^)]. Fig. 12. 11737—07 10 320 Bulleti7t of the Bureau of Standards. voi. Stock Photo$

RM2AM5B14–The self and mutual-inductance of linear conductors . -p LJ b log ^ |^H-|0 -4|^«log L ?b —yja^-^b-^b ] Putting ^a^--b^ — d^ the diagonal of the rectangle, 2ab . , 3<2 Z =: 4 <^ lop^- .+ 4 ^log- ^^ •^- 44 -^+/^] <f—^ —«+p or p{b-{-d)Z = 4 (<2-[-^)log a log (a-{-d) — b log (/^+(3^) 319 (24) For ^ = 200 cm, b= 100, /3 = o.i ^ = 8017.1 cm = 8.017 microhenrys.((^) 772^ conductor having a rectangular sectio7i. For a rectangle made up of a conductor of rectangular section, aX/S, A=2 r 1 2^ ; a log- ; ?^H-^+o.2235(a+^)]. Fig. 12. 11737—07 10 320 Bulleti7t of the Bureau of Standards. voi.

$The self and mutual-inductance of linear conductors . m. d. and a. m. s. d. inthe third term; that is, a term of the form Si log R, To get this we must integrate as follows: .^[log(.-.)-g+flog.-g (46) I f & I f 6 d^S^^ log i^2 = - I (^ —-;i:)^log {d—x)dx-r-^ I x^ log xdx I (b — xYdx I x^dx 9J0 9J0 6V ^ 12; (47) i^ogb-1^^ (48) ^/logi?, = |, We may now substitute in (39) as follows: -?log<3^=log^—- ^ d^— —^ 2 3(3^^ log <^=—(log ^—^ j 330 This gives Bulletin of the Bureau of Standards. ivoi. 4, no. 2^ L — AfiTa— /.ira I^ + 3&) ^°^ ^^-^^ ^-h (°^ -^2) -^-9&] A 32«/ 1 8<3; log- -^ ^ b —1 Stock Photo$

RM2AM5AP2–The self and mutual-inductance of linear conductors . m. d. and a. m. s. d. inthe third term; that is, a term of the form Si log R, To get this we must integrate as follows: .^[log(.-.)-g+flog.-g (46) I f & I f 6 d^S^^ log i^2 = - I (^ —-;i:)^log {d—x)dx-r-^ I x^ log xdx I (b — xYdx I x^dx 9J0 9J0 6V ^ 12; (47) i^ogb-1^^ (48) ^/logi?, = |, We may now substitute in (39) as follows: -?log<3^=log^—- ^ d^— —^ 2 3(3^^ log <^=—(log ^—^ j 330 This gives Bulletin of the Bureau of Standards. ivoi. 4, no. 2^ L — AfiTa— /.ira I^ + 3&) ^°^ ^^-^^ ^-h (°^ -^2) -^-9&] A 32«/ 1 8<3; log- -^ ^ b —1