EXACT CLOSED FORM FORMULA FOR SELF INDUC- TANCE OF CONDUCTOR OF RECTANGULAR CROSS SECTION

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1 Progress In Eectromagnetics Research M, Vo. 26, , 22 EXACT COSED FORM FORMUA FOR SEF INDUC- TANCE OF CONDUCTOR OF RECTANGUAR CROSS SECTION Z. Piatek, * and B. Baron 2 Czestochowa University of Technoogy, Brzeznicka 6A, Czestochowa 42-2, Poand 2 The Siesian University of Technoogy, Akademicka, Giwice 44-, Poand Abstract In this paper, sef inductance for a conductor with rectanguar cross section is investigated. Using the three-dimensiona Fredhom s integra equation of the second kind with weaky singuar kerne we obtain an equation for the compex votage drop in the conductor. Sef impedance appearing in the equation is expressed in the form of integra reation for any current density distribution. The imaginary part of this impedance divided by anguar frequency is the sef inductance of a conductor of any shape and finite ength. In the case of direct current DC, ow frequency F or thin strip conductor of rectanguar cross section the formuae for the sef inductances are given for any ength and for ength much greater than the other dimensions.. INTRODUCTION The rea umped isoated conductor can be modeed as a connection, in series or in parae, of a resistance and a sef inductance. Sef inductance pays an important roe not ony in power circuits, but aso in transmission ines, interconnections in many microwave and digita printed circuit board PCB ands and stripines ]. Formuae for sef inductances of conductors of rectanguar crosssection are the subject of many eectrica papers and books. As for DC, power frequency or a thin isoated rectanguar conductor with width a, thickness b, and ength as shown in Fig., there are many formuae for its sef inductance. The most significant are: Grover s Received 3 August 22, Accepted 2 September 22, Schedued 22 October 22 * Corresponding author: Zygmunt Piatek zygmunt.piatek@interia.p.

2 226 Piatek and Baron given in 2, 3, 2, 3], Bueno and Assis shown in ], FastHenry s introduced in 3], Ruehi s presented in 3,, 3] as we as Hoer and ove s shown in 3, 3, 4]. In genera cases, there are two methods to cacuate sef inductance: the first one is the cacuation of inductance of a currentcarrying cosed oop and the second is the cacuation of induction of a segment of a current oop using the concept of partia inductance 3]. In this paper, a new method for cacuating sef inductance is presented. The method resuts in the Fredhom s integra equation. We compare our formuae with severa we-known ones given in the iterature. 2. INTEGRA EQUATION In case of a rectiinear conductor with ength, conductivity σ, cross section S with sinusoida current input function with anguar frequency ω and compex vaue I Fig., the vector of current density has one component aong the Oz axis, that is J X = a z J X. Then the vector magnetic potentia A X = a z A X and tota eectric fied E X = a z E X. The tota eectric fied may be presented 5 7] in the form of two eectric fieds sum E X = E st X + E in X where E st X is the so-caed quasi-static source whist E in X is an induced eectric fied. y a E Xx, y,z a z a y a x b x J ρ XY ϕ I Yx, y,z ϕ z 2 ϕ z z ϕ Figure. A conductor of rectanguar cross section with width a, thickness b, ength, conductivity σ and compex current I.

3 Progress In Eectromagnetics Research M, Vo. 26, The quasi-static source eectric fied E st X = gradϕz 2 where ϕz is the compex scaar eectric potentia aong a straight conductor of rectanguar cross section of ength Fig.. This potentia must satisfy apace s equation d 2 ϕ z dz 2 = 3 the soution of which is 5 7] ϕ ϕ ϕ z = z + ϕ 4 By introducing a unit votage drop in V m in a conductor ϕ ϕ u = 5 we obtain from the formua 4 ϕ z = u z + ϕ 6 then from it and from the formua 2 E st X = dϕz = u 7 dz The induced eectric fied E in X is determined by E in X = jωa X, where A X is a vector magnetic potentia and A X = µ J Y dx 2 dy 2 dz 2 8 4π ρ XY v where: X = Xx, y, z is the point of observation and X R 3, Y = Y x 2, y 2, z 2 is the source point and Y S, v conductor s voume, ρ XY = rxy 2 + z 2 z 2 is the distance between the point of observation X and the source point Y Fig., where r XY = x2 x 2 + y 2 y 2. Finay becomes E X + j ωa X = u 9 and from it, after substituting E X = JX σ and A X from 8, for X v we have J X + jω µ J Y dx 2 dy 2 dz 2 = u σ 4π ρ XY v Equation is a three-dimensiona Fredhom s integra equation of a second type with a weak-singuar kerne ρ XY and it has an unequivoca soution for any not ony constant right side 6, 7].

4 228 Piatek and Baron 3. DEFINITION OF SEF INDUCTANCE We mutipy by the compex conjugate vaue J X and integrate over the voume v of this conductor. Assuming that J X does not depend on the variabe z and taking into consideration that U = u is the votage drop in the conductor in V as we as that I 2 = I I, we may divide both sides of the fina equation by I. Then we obtain the equation U = Z I = R + j ω I where the sef-inductance of the conductor = ω Im Z = µ J Y J X 4πI 2 dv dv 2 ρ XY v One can see from 2 that the sef inductance of the conductor depends on the distribution of the current density in this conductor. The formua 2 specifies the sef-inductance of a soid conductor as a parameter standing next to jω in which specifies the votage drop and from the point of theory of circuits it is caed 6, 7] sef inductance. It may not be associated with a cosed oop according to the cassica view of sef-inductance of a cosed circuit but it shoud be merey considered as a quantity hepfu in cacuating sef inductances of rea cosed eectrica circuits. The above formua can be aso deduced from the magnetic energy of a oop carrying current I 5]. If a conductor has a constant cross-sectiona area S aong its ength and in the case of DC, ow frequency or for a thin strip conductor we can assume that the current density is uniform and given as JX = I /S and then, from 2, we obtain the sef inductance of a straight conductor = µ 4πS 2 v v v ρ XY dv dv 3 4. SEF INDUCTANCE OF A CONDUCTOR OF RECTANGUAR CROSS SECTION The sef inductance of a rectanguar conductor of dimensions a b shown in Fig. is given by the formua = µ 4π a 2 b 2 F 4 where F = b b a a ρ XY dx dx 2 dy dy 2 dz dz 2 5

5 Progress In Eectromagnetics Research M, Vo. 26, is a sixtupe definite integra of an integrabe function ρ XY of six variabes x, x 2, y, y 2, z, z 2. In genera cases this integra is very difficut to cacuate. But in our case we can put x = x 2 x, y = y 2 y, z = z 2 z and first cacuate a sixtupe indefinite integra F x, y, z = dx dx dy dy dz dz 6 ρ XY x, y, z of a function ρ XY x, y, z = x 2 + y 2 + z of three variabes x, y, z twice with respect to x, twice with respect to y and twice with respect to z. After each doube integration we omit terms which depend ony on one or two variabes they are constants with respect to the considered variabe. If two variabes, for exampe x and x 2, can be repaced with ony one variabe x = x 2 x then a doube definite integra can be cacuated from foowing formua where H = s 4 s 2 fx 2 x, y, zdx 2 dx s 3 s = Hs 4 s Hs 4 s 2 + Hs 3 s 2 Hs 3 s s 4 s,s 3 s 2 p, p 3 i=4 = Hx] x = Hx] x = i+ Hp k 8 s 4 s 2,s 3 s p 2,p 4 Hx, y, z = i= fx, y, zdx dx 9 is an indefinite integra of fx, y, z. So in 6, we can aso omit terms proportiona to one variabe ike Hx, y, z = xgy, z. Finay, after a engthy integration, 6 yieds an expression for the sixtupe indefinite integra F x, y, z = dx dx dy dy dz dz ρ XY x, y, z = dx dx dy dy dz dz x 2 + y 2 + z2

6 23 Piatek and Baron 6 5 x 4 +y 4 +z 4 3x 2 y 2 3x 2 z 2 3y 2 z 2 x 2 +y 2 +z 2 2xyz z 2 tan xy z x 2 +y 2 +z 2 = +y 2 tan xz + y x 2 +y 2 +z x2 tan yz 2 x x 2 +y 2 +z xy 4 6y 2 z 2 + z 4 n x + x 2 + y 2 + z 2 3yx 4 6x 2 z 2 + z 4 n y + x 2 + y 2 + z 2 3zx 4 6x 2 y 2 + y 4 n z + x 2 + y 2 + z 2 2 and the sef inductance of the conductor of rectanguar cross section is given by foowing formua ] ] = µ a, a b, - b, 4π a 2 b 2 F x, y, z] x y z,,, = µ 4π a 2 b 2 i=4 j=4 k=4 i+j+k+ F p i, q j, r k 2 i= j= k= where p = a, p 3 = a, q = b, q 3 = b, r =, r 3 = and p 2 = p 4 = q 2 = q 4 = r 2 = r 4 =. On the basis of 2, we have an anaytica formua for the sef inductance of the straight conductor of rectanguar cross section µ = 2πa 2 b 2 4 a 5 + b a 4 3a 2 b 2 + b 4 a 2 + b 2 4 a 4 3a a b 4 3b b a 4 + b a 2 b 2 3a 2 2 3b 2 2 a 2 + b ab a 2 tan b a a 2 +b b2 tan a b a 2 +b tan +3ab ab n + a 2 +b b+ +a n a 2 +b a 2 +b 2 +2 b+ a+ + b n a 2 +b µ 2πa2 b 2 5a b 4 n a+ a 2 +b 2 a+ a 2 +b a+ a 2 +b 2 a+ a 2 +b n a+ 5b a 4 n b+ a 2 +b 2 b+ a 2 +b b+ a 2 +b 2 b+ a 2 +b n b+ 5 a 4 n + a a 2 +b a a 2 +b b4 n + ] ab a 2 +b a 2 +b a+ a 2 +b ] a a+ a 2 +b a+ a a+ a 2 +b b b+ a 2 +b b+ b b+ a 2 +b b ] a 2 +b b a 2 +b ] ] 22

7 Progress In Eectromagnetics Research M, Vo. 26, SEF INDUCTANCE OF A THIN TAPE The sef inductance of a thin tape of width a, thickness b and ength is given by formua where F = a a = µ 4π a 2 F 23 dx dx 2 dz dz 2 x2 x 2 + z 2 z 2 24 is a quadrupe definite integra of four variabes x, x 2, z, z 2. Now we can put x = x 2 x and z = z 2 z and first cacuate a quadrupe indefinite integra dx dx dz dz F x, z = 25 x 2 + z 2 twice with respect to x and twice with respect to z. Finay, after a engthy integration, 25 yieds an expression for quadrupe indefinite integra F x, z = z x 2 + z 2 x 2 + z 2 3/2 3 +xz 2 n x + x 2 + z 2 + x 2 z n z + ] x 2 + z 2 26 So the sef inductance of the thin tape is given by the foowing formua ] = µ, 4π a 2 F x, z] = µ i=4 k=4 4π a 2 i+k F p i, r k 27 a, a x, z, i= k= On the basis of 27 we have an anaytica formua for the sef inductance of the thin tape = µ 6πa 2 3a 2 n a a 2 3/2 a +3a 2 n a a a 3 ] It is exacty the Hoer s formua for the thin tape given in 4]. 28

8 232 Piatek and Baron 6. CASE OF ENGTH OF CONDUCTOR MUCH GREATER THAN THE OTHER DIMENSIONS The doube definite integra is fx, y= dz dz 2 =2 n + 2 +rxy 2 2 +rxy 2 + r XY 29 ρ XY r XY If r XY the function fx, y becomes fx, y = 2 n 2 r XY 3 and the sef inductance of the rectanguar conductor is expressed by the formua = µ n 2 + G] 3 2π where G = b b a a 2a 2 b 2 n x 2 x 2 + y 2 y 2] dx dx 2 dy dy 2 32 After cacuating quadrupe indefinite integra Gx, y = 2a 2 b 2 n x 2 + y 2] dx dx dy dy { = 288a 2 b 2 5x 2 y 2 6 8xy 3 tan x y + 8x3 y tan y x x 4 6x 2 y 2 + y 4 n x 2 + y 2 ]} 33 we determine the sef inductance of the ong conductor of rectanguar cross section { ] } = µ b, - b n 2 + Gx, y] 2π = µ 2π a, a x, y, i=4 n 2 + j=4 i+j Gp i, q j i= j= 34 On the basis of 34 we have the anaytica formuae for the sef

9 Progress In Eectromagnetics Research M, Vo. 26, inductance of the straight ong conductor of rectanguar cross section = µ { n 2 2π a b a 3 a tan b + a b ] b tan a + a ] 2 6+ b 2 a ] 2 n + + a ] } 2 6 n a 35 2 b a b 6 b b or = µ 2π + 2 { n 2 b b a 3 a tan b + a b ] b tan a a ] 2 6+ b 2 ] b 2 n b a a 6 b a 2 ] n b a } 36 as we as { = µ n 2 2π a+b ] b a 3 a tan b + a b ]+ 2 b tan a n + a 2 b + a 2 b + a ] 2 b 2 b 2 a ] ]} 2 n + + n b a a b 7. COMPUTATIONA RESUTS For fast digita computations the foowing normaizations are introduced: u = /a and w = b/a. From 22 the per-unit-ength sef inductance of the straight conductor of rectanguar cross section is then + w3 uw 2 u + u4 3 w 2 uw 2 u + w2 u A u + u3 A uw 2 w 2 w 2 w 2 u3 u 3u + A w w2 uw 2 u + u3 3 w 2 u 3 u 3u A w 2 4 = µ π 3 w tan uw A 4 + w tan n u+a 4 u wa 4 + u2 w tan u+a 4 + u w n w+a 4 w+a 4 + u n +A 4 +A 4 w ua 4 w 2 u n +A 2+A 4 +A 2 +A 4 + u3 n +A +A 4 w 2 +A +A 4 + uw n w+a 2w+A 4 w n w+a 3w+A 4 w+a 2 w+a 4 + u3 w+a 3 w+a 4 + n u+a u+a 4 w 2 u+a u+a 4 + w2 n u+a 3u+A 4 u+a 3 u+a 4 38 where A = + u 2, A 2 = + w 2, A 3 = u 2 + w 2 and A 4 = + u 2 + w 2.

10 234 Piatek and Baron From 28 the per-unit-ength sef inductance of the thin tape is given by foowing formua = µ 3 n u+ +u 6π 2 +u 2 3/2 + ] +u 2 +3u n +u u u u From 35, 36 and 37 the per-unit-ength sef inductance of the straight ong conductor of rectanguar cross section is given by foowing formuae or = µ 2π as we as + 2 n 2u w tan 3 w + w tan w n + w w w2 6 w 2 n w = µ n 2u 2π w w tan 3 w + w tan w + 2 w w2 n + w w 2 ] n w 6 = µ 2π { n 2u + w n + 2w +w 2 ] w tan 3 w + w tan w + 2 w 2 n +w 2 +w 2 n + w ]} 2 42 For given rations u = /a and w = b/a the per-unit-ength sef inductance is independent of the vaues of width a and thickness b. Tabe. Per-unit-ength sef inductance of a rectanguar conductor for normaized ength for DC or ow frequency. u = /a Grover nhm Bueno FastHenry nhm nhm w = b/a =. Ruehi Hoer Eq. 38 Eq. 39 Eq. 4 nhm nhm nhm nhm nhm. negative negative negative. negative negative negative *According to Eq. 2 from 4].

11 Progress In Eectromagnetics Research M, Vo. 26, For the chosen ratio w = b/a and different normaized engths of the straight conductor of rectanguar cross section the cacuations of its per-unit-ength sef inductance have been made according to a previous, shown above, formuae Tabe. 8. CONCUSIONS In this paper, we have presented a new method for cacuating sef inductance of an isoated conductor of rectanguar cross section. We have defined the sef inductance of conductor of any shape and finite ength given by sixtupe definite integra. In the case of DC or ow frequency we have given genera formuae for sef inductance of conductors of rectanguar cross section of any dimensions incuding thin tapes and very ong conductors. By computations we have shown that our formuae give the same resuts as FastHenry s, Ruehi s and a Hoer s formuae 2 and 22 for a dimensions of a conductor. But computationa resuts show that our formuae are numericay more stabe and accurate for a dimensions of a conductor, particuary for ong on-chip interconnections, than the others. In addition we have aso obtained anaytica forms of a formuae, which are more usefu than genera ones. Of course they give the same resuts as the genera formuae. Our formuae are anayticay simpe and can aso repace the traditiona working ones or tabes. These formuae can be used in the methods of numerica cacuation of AC sef inductance of a rectanguar conductor. Then the cross section of the conductor is divided into rectanguar subbars eementary bars in which the current is assumed to be uniformy distributed over the cross section of each subbar. ACKNOWEDGMENT This work was supported by the Nationa Science Centre, Poand, under Grant No. N N REFERENCES. Bazer, G., et a., Switcher Manua, th edition, ABB Caor Emag Mittespannung GmbH, Ratingen, Kazimierczuk, M. K., High-frequency Magnetic Components, J. Wiey & Sons, Chichester, Pau, C. R., Inductance: oop and Partia, J. Wiey & Sons, New Jersey, Antonini, G., A. Orandi, and C. R. Pau, Interna impedance of conductor of rectanguar cross section, IEEE Trans. Microwave Theory Tech., Vo. 47, No. 7, , 999.

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