Holloway rectangular thing

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY 1 DC Internal Inductance for a Conductor of Rectangular Cross Section Christopher L. Holloway , Senior Member, IEEE , and Edward F. Kuester , Fellow, IEEE Abstract —In this paper, the dc internal inductance for conduc- tors with rectangular cross section is investigated. Using a qua- sistatic Green’s function, the magnetic fields inside a rectangular conductor are derived. These magnetic field expressions are used to formulate the internal inductance of the conductor. We present numerical results and closed-form expressions for the dc internal inductance for this geometry. Comparisons to various expressions found in the literature are made, and the inaccuracies of these other results are presented and discussed. Finally, we discuss some of the subtleties associated with the inductance of a single current path, in which we examine why the total and external inductances of a single infinitely long conductor have no physical meaning, whereas the internal inductance does. Index Terms —DC internal inductance, interconnect, printed circuit broads (PCB), rectangular conductor, signal integrity, transmission lines. I. I NTRODUCTION IT IS WELL known [1] that the inductance per unit length of a circular cylindrical conducting wire is dependent on frequency. At dc, its inductance includes a contribution from the magnetic energy stored within the conductor itself, in addition to that stored in the field outside the wire. As frequency increases, the skin effect begins to reduce the internal field, until in the limit of infinite frequency, the part of the inductance due to this field (the internal inductance ) vanishes. A wide array of planar transmission line structures (mi- crostrip line, stripline, coplanar waveguides (CPW), coplanar strips (CPS), etc.) used for interconnects and high-speed printed circuit broads (PCBs) have signal traces with rectangular rather than circular cross sections. In the design of these types of trans- mission line structures, it is important to consider the waveform distortion. This is especially important in signal integrity con- siderations (see [2]–[6]). In such analyses, frequency-dependent models are used to investigate high-speed digital waveforms that propagate along lossy and dispersive transmission lines. The fre- quency dependencies of the conductor resistance and inductance caused by the skin effect are particularly important [7]–[27]. In these dispersive line models, knowledge of the dc and infinite frequency limits of resistance and inductance are needed to ob- tain the correct functional forms of the frequency dependence for transmission line parameters. Manuscript received December 9, 2008. C. L. Holloway is with the Electromagnetics Division, National Institute of Standards and Technology, U.S. Department of Commerce, Boulder Laborato- ries, Boulder, CO 80305 USA (e-mail: [email protected]). E. F. Kuester is with the Department of Electrical and Computer Engineer- ing, University of Colorado, Boulder, CO 80309-0425 USA (e-mail: edward. [email protected]). Digital Object Identifier 10.1109/TEMC.2009.2016104Of interest in this paper is the 2-D rectangular cross-sectional conductor illustrated in Fig. 1. The conductor is infinitely long in the zdirection and is centered at the origin (i.e., y=0 and x=0). The conductor is assumed to have a thickness tand a width w. The dc internal resistance (per unit length or pul)f o r such a rectangular conductor is well known and given by Rdc=1 σwt(1) where σis the conductivity of the conductor. The dc inductance of this conductor is less well known, and although a number of papers have given expressions for this structure [28]–[32], they are not accurate, nor do they agree with one another. Moreover, in contrast with the case of a circular wire, the less symmetrical rectangular conductor’s change in inductance with frequency cannot be exclusively attributed to the reduction in internal inductance. As frequency increases, the fields external to the rectangular conductor will change to some extent, meaning that the entire inductance must be considered for practical applica- tions, not just the internal part. We will treat the change in the inductance with frequency (from dc to high-frequency limit) for rectangular conductors in a future paper. In this current paper, we concentrate on the internal dc inductance Li. Here, we will present accurate results for the dc internal in- ductance of a rectangular cross-sectional conductor. In obtain- ing these results, we first derive expressions for the magnetic fields inside a rectangular conductor that are obtained from a quasistatic Green’s function approach. These expressions are used to calculate the stored energy (and, in turn, the internal inductance) of this geometry. We present two approximate ex- pressions for the dc internal inductance, one that is valid if t/w < 0.01, and a second one valid for 0.01 <t / w< 1.W e will need to discuss results only for t/w≤1, because of the symmetry of the cross section. The paper is organized as follows: after the introduction, Section II presents the Green’s function approach and illus- trates how it is used to obtain the magnetic fields inside the conductor. This section also lays out the formulation of the dc internal inductance. In a section, we present an approximation for the magnetic fields for small t/w that is used to obtain a closed form for the internal inductance in this limit. For arbi- trary values of t/w, the internal inductance requires a numerical evaluation of an integral of the magnetic fields, which is dis- cussed in Section III. In this section, we also present an approx- imate expression for the dc internal inductance for an arbitrary value of t/w. Inaccuracies in other results found in the literature are also discussed. Section VI summarizes the results presented here. Finally, in the Appendix, we discuss some of the subtleties associated with the inductance of a single current path, in which 0018-9375/$25.00 © 2009 IEEE 2 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY Fig. 1. Conductor with a rectangular cross section. we examine why the total and external inductance of a single infinitely long conductor have no physical meaning, whereas the internal inductance does. II. F ORMULATION The internal inductance of a conductor of arbitrary shape is given by [1] Li=1 I2/integraldisplay ¯B·¯Hd V (2) where BandHare the magnetic field and magnetic field inten- sity inside the conductor, and the integral is carried out over the entire conductor volume. If the conductor is straight, has length land a uniform cross section along the z-axis, then BandH become independent of zin the limit as l→∞ , and the internal inductance per unit length can be expressed as Li l=1 µI2/integraldisplay ¯B·¯Bd S (3) where the integral is now carried out over the cross section of the conductor, and it has been assumed that µis constant throughout the conductor. One has to be careful with the interpretation of inductances of an infinitely long single conductor, so a few comments are needed about the inductance. In the Appendix, it is shown that the internal inductance for a single isolated infinite conductor does have meaning, even though its external inductance is un- defined. The problem arises from the logarithmic divergence of the magnetic potential Awhenl→∞ ,a sd i s c u s s e di nt h e Appendix. It is shown in the Appendix that (3) is finite. For an arbitrarily shaped straight conductor with current flow- ing along the z-axis, the B-fields can be obtained from the z- component of the magnetic vector potential ¯A, where it can be shown that [1] Bx=∂ ∂yAzandBy=−∂ ∂xAz. (4) The magnetic vector potential for a long current-carrying con- ductor can be expressed as (see the Appendix) Az=−µ 4π/integraldisplay Jln[(x−x/prime)2+(y−y/prime)2]dx/primedy/prime+C (5) where Jis the current density in the conductor, and Cis a constant. From (4), we see that the derivatives of this constant give zero contribution to the Bfields.For the rectangular cross-section conductor shown in Fig. 1 with a uniform dc current density given by J=I/(wt),t h e magnetic vector potential is expressed as Az=−µI 4πwt/integraldisplayt/2 −t/2/integraldisplayw/ 2 −w/ 2ln[(x−x/prime)2 +(y−y/prime)2]dx/primedy/prime+C. (6) Strutt [33], [34] evaluated this integral to obtain the magnetic vector potential (to within a constant) for the rectangular con- ductor. Strutt’s results can also be found in [35] and [36]. His result for Azcould be substituted into (4) to obtain expressions for the Bfield. An alternative approach is to first recognize that the derivatives in (4) are in terms of xandy, while the integral in (6) is over y/primeandx/prime. As such, the derivatives with respect to xandycan be moved inside the integral and carried out analyt- ically before evaluating the integrals. This allows one to obtain theBfields directly by evaluating the following integrals: Bx=µI 2πwt/integraldisplayt/2 −t/2/integraldisplayw/ 2 −w/ 2y/prime−y (x/prime−x)2+(y/prime−y)2dx/primedy/prime (7) and By=−µI 2πwt/integraldisplayt/2 −t/2/integraldisplayw/ 2 −w/ 2x/prime−x (x/prime−x)2+(y/prime−y)2dx/primedy/prime. (8) Note that the constant Cis eliminated. These two double integrals are fairly straightforward to eval- uate, and leaving out the details, it can be shown that the x- component of the Bfield is given by Bx=µI 2πwtW1 (9) and the y-component by By=−µI 2πwtW2 (10) where W1=/parenleftbiggw+2x 4/parenrightbigg ln/bracketleftbigg(w/2+x)2+(t/2−y)2 (w/2+x)2+(t/2+y)2/bracketrightbigg +/parenleftbiggw−2x 4/parenrightbigg ln/bracketleftbigg(w/2−x)2+(t/2−y)2 (w/2−x)2+(t/2+y)2/bracketrightbigg +(t/2−y)/bracketleftbigg tan−1/parenleftbiggw−2x t−2y/parenrightbigg + tan−1/parenleftbiggw+2x t−2y/parenrightbigg/bracketrightbigg −(t/2+y)/bracketleftbigg tan−1/parenleftbiggw−2x t+2y/parenrightbigg + tan−1/parenleftbiggw+2x t+2y/parenrightbigg/bracketrightbigg (11) and W2=/parenleftbiggt+2y 4/parenrightbigg ln/bracketleftbigg(w/2−x)2+(t/2+y)2 (w/2+x)2+(t/2+y)2/bracketrightbigg +/parenleftbiggt−2y 4/parenrightbigg ln/bracketleftbigg(w/2−x)2+(t/2−y)2 (w/2+x)2+(t/2−y)2/bracketrightbigg HOLLOW AY AND KUESTER: DC INTERNAL INDUCTANCE FOR A CONDUCTOR 3 +(w/2−x)/bracketleftbigg tan−1/parenleftbiggt−2y w−2x/parenrightbigg + tan−1/parenleftbiggt+2y w−2x/parenrightbigg/bracketrightbigg −(w/2+x)/bracketleftbigg tan−1/parenleftbiggt−2y w+2x/parenrightbigg + tan−1/parenleftbiggt+2y w+2x/parenrightbigg/bracketrightbigg . (12) This approach was used by Hague [35], but his results are incorrect. With the xandycomponents of the Bfields in hand, (3) can be used to obtain the following for the dc internal inductance per unit length of the rectangular conductor Li l=µ (2πwt )2/integraldisplayt/2 −t/2/integraldisplayw/ 2 −w/ 2/bracketleftbig W2 1+W2 2/bracketrightbig dx dy. (13) Unfortunately, these integrals cannot be evaluated in closed form. The problem is that the integrand contains terms such as [tan−1(f(x,y )/g(x,y )]2and ln2[v(x,y )/w(x,y )]. These types of integrals result in functions known as polylogarithms [37], which cannot, in general, be expressed in terms of elementary functions. More importantly, the cross terms result in integrals that cannot be evaluated even in terms of standard special func- tions. In Section III, we present a numerical evaluation of this integral. Along with this numerical evaluation, we will also give approximate formulas for Li/lfor arbitrary values of t/w.H o w - ever, for small t/w, a closed-form result can be obtained. This is the topic of the next section. A. Internal Inductance for Small t/w From (9)–(12), it can be shown that for small t/w, the fields can be expressed as Bx=−µI 4πwW3 (14) while the y-component is expressed as By=µI 4πwW4 (15) where W3=4yπ t+/parenleftbigg 1−2y t/parenrightbigg/bracketleftbigg tan−1/parenleftbiggt−2y w−2x/parenrightbigg + tan−1/parenleftbiggt−2y w+2x/parenrightbigg/bracketrightbigg −/parenleftbigg 1+2y t/parenrightbigg/bracketleftbigg tan−1/parenleftbiggt+2y w−2x/parenrightbigg + tan−1/parenleftbiggt+2y w+2x/parenrightbigg/bracketrightbigg (16) and W4=l n/parenleftbigg(w/2−x)2+y2 (w/2+x)2+y2/parenrightbigg . (17) If these fields are squared and substituted in (3), the internal inductance for small t/w is given by Li l=µ (4πw)2/integraldisplayt/2 −t/2/integraldisplayw/ 2 −w/ 2/parenleftbig W2 3+W2 4/parenrightbig dx dy. (18) Unfortunately, we have the same problem as earlier; that is, the squared arctangents and logarithms result in integrals that cannotbe evaluated in closed form. However, by making further small- argument approximations for the logarithms and arctangents in (16) and (17), the integral can be evaluated in closed form. We start with W3. It can be shown that by use of a Taylor series expansion of the arctangents for small (t/2±y)/(w/2±x), W3can be approximated to first order by W3≈2yπ t/2−4y/bracketleftbigg1 w/2−x+1 w/2+x/bracketrightbigg . (19) The first term dominates, and thus /integraldisplayt/2 −t/2/integraldisplayw/ 2 −w/ 2W2 3dx dy≈4 3π2wt. (20) For small (y)/(a±x),the logarithm in W4can be approximated to first order as W4≈2/bracketleftbigg ln/parenleftbiggw/2−x w/2+x/parenrightbigg/bracketrightbigg +y2 (w/2−x)2−y2 (w/2+x)2. (21) The first term dominates once again; thus /integraldisplayt/2 −t/2/integraldisplayw/ 2 −w/ 2W2 4dx dy≈4t/integraldisplayw/ 2 −w/ 2/bracketleftbigg ln/parenleftbiggw/2−x w/2+x/parenrightbigg/bracketrightbigg2 dx. (22) This integral is evaluated to be /integraldisplayw/ 2 −w/ 2/bracketleftbigg ln/parenleftbiggw/2−x w/2+x/parenrightbigg/bracketrightbigg2 dx=w 3π2(23) and so we have /integraldisplayt −t/2/integraldisplayw/ 2 −w/ 2W2 4dx dy≈4 3π2wt. (24) Substituting (20) and (24) into (18), the dc internal inductance for small t/w can be approximated by Li l≈µ 6t w. (25) It is interesting to note that for the small t/w ratio, the Bxand Byfields contribute identical amounts to the stored energy in the conductor. In the next section, this approximate expression for small t/w is compared with a numerical evaluation of the integral in (13). III. DC I NTERNAL INDUCTANCE :RESULTS AND COMPARISONS The functions W2 1andW2 2are well-behaved functions, and thus, a numerical evaluation of the integration given in (13) is straightforward. This numerical integration can then be used to obtain Li/lfor any value of t/w. Figs. 2 and 3 show the results for the dc internal inductance Li/lobtained from this numeri- cal integration. The results in Fig. 3 are on a log scale in order to see more clearly the behavior for small values of t/w.T h e results are obtained assuming µ=µ0. In both these figures, we show results only for t/w≤1. The rectangular cross section is a symmetric geometry; thus, the internal inductance is sym- metric about t/w =1 [in the sense that Li(t/w )=Li(w/t )], which is where the inductance reaches its maximum. As a com- parison, the dc internal inductance of a circular wire (which is 4 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY Fig. 2. DC internal inductance of a rectangular conductor as a function of t/w. Fig. 3. Log plot of the dc internal inductance of a rectangular conductor as a function of t/w. µ0/8π=5 0 nH/m) is shown in the figures, while for a square conductor, the internal inductance is found to be 48.3 nH/m. In these figures, we also show the results obtained from the thin- strip approximation, t/w/lessmuch1(25). From these figures, we see that for t/w < 0.01, the expression in (25) is indistinguishable from that of the numerical integration of (13).There are only a few results in the literature for the dc in- ternal inductance of a rectangular cross-section conductor; see [28]–[32]. Antonini et al. [31] present results of the frequency- dependent internal resistance and inductance obtained from a full numerical approach of subdividing the conductor cross sec- tion. In their paper, they investigate the high-frequency depen- dence; however, they also present a dc result for a square cross section. They state that the value is on the order of 48 nH/m. Our results give an internal inductance for a square cross section of 48.3 nH/m. An approximate formula for the internal inductance can be found in [28]. Using a variational approach, Kalantarov and Tseitlin [28] derived the following expression for the dc internal inductance: Li,KT l=0.221µ πwt w2+t2. (26) This expression is plotted in Figs. 2 and 3. From this comparison, we see that this expression is in error by about 8.5%fort/w =1. The error increases quickly as t/w decreases, rising to 82% at t/w =0.1. Tsiboukis and Kriezis [29] also used a variational approach along with the Rayleigh–Ritz method to obtain the following approximate formula: Li,TK l=0.209µ π∞/summationdisplay n=1,3,...∞/summationdisplay m=1,3,...wt n2m2(n2w2+m2t2). (27) This expression is somewhat cumbersome because it requires an infinite summation. However, each summation converges after ten terms (i.e., the values are indistinguishable for ten terms when compared to the values for ten thousand terms in the summations). When converged, the results from this expression give results very similar to those of Kalantarov and Tseitlin, as seen in Fig. 2. The results from (27) in this figure were obtained using ten thousand terms in each summation. Hence, the error of (27) is essentially the same as that of Kalantarov and Tseitlin’s expression. Krakowski and Morawska [30] analyzed the skin effect of thin conductors (i.e., t/w/lessmuch1). In their analysis, they obtained the following for the dc internal inductance: Li,KM l=µ 6w t. (28) This expression is obviously incorrect, since it predicts a non- physical result of increasing Liastdecreases. The problems with the derivation in [30] are twofold. First of all, taking the limit as frequency goes to zero in (9) of [30] is problematic, and results in very large field values for small t. In fact, we showed earlier that for small t/w,Byvaries as 1/wand not as 1/t,a s predicted in [30]. Second, in [30], only the Hycomponent was used in calculating the stored energy. From (14) and (15), we see that both fields are present, and from (20) and (24), it is seen that for low aspect ratios t/w, both the xandymagnetic field components contribute the same amount toward the stored energy. HOLLOW AY AND KUESTER: DC INTERNAL INDUCTANCE FOR A CONDUCTOR 5 Fig. 4. Comparison of proposed approximate formulas for the dc internal inductance of a rectangular conductor as a function of t/w. Finally, Chen and Fang [32] present the following polynomial approximate formula for the dc internal inductance: Li,CF l=µ 8π[1.07143−0.08608( w/t−1) +0.00553( w/t−1)2−1.99 ×10−4(w/t−1)3+2.863×10−6(w/t−1)4]. (29) The claim in [32] is that this expression is valid for 1≤w/t≤ 20(or0.05≤t/w≤1). The expression is plotted in Figs. 2 and 3. The first thing we notice is that for t/w =1, this expression gives a value of 53.6 nH/m, which is larger than both our result and that of Antonini et al. [31]. Indeed, the value predicted from (29) is also larger than for that for a circular wire (50 nH/m). Notice also that the expression has significant error for all values oft/w. While our results in Figs. 2 and 3 give accurate results for the dc internal inductance, they are not in closed form. By assuming that t/w << 1, an accurate closed-form result was obtained for t/w < 0.01in (25). A closed-form expression valid for 0.01<t / w< 1can also be obtained by approximating the numerical results in Figs. 2 and 3 with a polynomial fit. Such a fit gives the following expression for the dc internal inductance in units of nH/m: Li,fit l[nH/m ]=0.045 + 200 .23(t/w )−443.01(t/w )2 + 724 .18(t/w )3−823.70(t/w )4 + 537 .40(t/w )5−146.845(t/w )6, for 0.01<t / w< 1. (30) Fig. 5. Log plot of the comparison of proposed approximate formulas for the dc internal inductance of a rectangular conductor as a function of t/w. Figs. 4 and 5 compare this expression to the results obtained from the numerical evaluation of the integral in (13). In fact, (30) exhibits a relative error of no larger than 0.1% for 0.005< t/w < 1. Therefore, the dc internal inductance is well approximated for all values of t/w by the following expressions: Ldc,i l[nH/m ]=Li,fit l, for 0.01<t / w ≤1 400π 6t w,fort/w≤0.01(31) where Li,fit/lis given in (30). From these comparisons, we see that these two expressions are indistinguishable from the numerical evaluation of the integral of (13) for all values of t/w. IV . C ONCLUSION In this paper, we investigated the dc internal inductance for conductors with rectangular cross sections. We have presented numerical results and closed-form expressions for the dc internal inductance for this geometry that are valid for any value of t/w. We gave two different expressions, one valid for t/w < 0.01, and a second for 0.01<t / w< 1. Values from these two ex- pressions are indistinguishable from numerical results. Com- parisons to other expressions found in the literature were made, and the inaccuracies of these other results were presented and discussed. In this paper, we also obtained expressions for the magnetic fields associated with a constant current flowing in the rectangular conductor. These expressions are valid both inside and outside the conductor. Finally, in the Appendix, we dis- cuss why the total and external inductance of a single infinitely 6 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY long conductor has no physical meaning, whereas the internal inductance does. APPENDIX INDUCTANCE OF A SINGLE CURRENT PATH As stated earlier, the internal inductance can be expressed in terms of the magnetic field ( B) and field intensity ( H). If we use (3) and recall that the Bfield can be expressed in term of the magnetic vector potential Aas ¯B=∇× ¯A (32) the internal inductance can be expressed as Li l=1 I2/integraldisplay ¯B·¯Hd S =1 I2/integraldisplay ∇× ¯A·¯Hd S . (33) With a few vector identities, we can show that Li l=1 I2/integraldisplay ∇× ¯A·¯Hd S =/integraldisplay/bracketleftbig (∇× ¯H)·¯A−∇ ·/parenleftbig¯HׯA/parenrightbig/bracketrightbig dS =/integraldisplay ¯J·¯Ad S−/contintegraldisplay/parenleftbig¯HׯA/parenrightbig ·¯andl. (34) The first term on the right-hand side of this expression can be “loosely” interpreted as the total inductance for a single conductor, while the second term can be “loosely” interpreted as the external inductance for a single conductor. We use the word “loosely” because for an infinitely long single conductor, these two terms are both singular (logarithmically infinite), but we will see that these two singularities cancel one another and give a finite and unique value for the internal inductance Li/l for a single conductor. The difficulties arise from ¯A. Starting with the static free- space Green’s function, it can be shown that the magnetic vector potential for a single conductor, whose length lis large compared to the cross-sectional dimensions, is given by Silvester [38] (we have corrected a sign error in his expression; see, for example, [39]) Az=Az,(x,y )+Az,C (35) where Az,(x,y )is the part of Azcontaining the xandydepen- dence, expressed as Az,(x,y )=−µI 4πSc/integraldisplay x/integraldisplay yln/bracketleftbig (x/prime−x)2+(y/prime−y)2/bracketrightbig dy/primedx/prime (36) whileAz,Cis a constant independent of xandy, given by Az,C =µI 2πln(2l). (37) Scis the cross-sectional area of the conductor. The singularity inAzarises from Az,C, and we see that as l→∞ ,Az,C→∞ . This singularity results in a singularity in the first and second terms of (34) (the total and external inductance of a single conductor, respectively). However, the singularities cancel when determining the internal inductance for a single conductor, i.e., when combining the external and total inductance. This is shownby the following. From (34), the total inductance of the single conductor is expressed as Ltotal =/integraldisplay ¯J·¯Ad S =/integraldisplay ¯J·¯A(x,y )dS+/integraldisplay ¯J·¯ACdS =/integraldisplay ¯J·¯A(x,y )dS+Az,CJSc.(38) We see that Ltotal→∞ asl→∞ (arising from Az,C) and Ltotal has no meaning for an infinitely long single conductor. From (34), the external inductance is Lex=/contintegraldisplay (¯HׯA)·¯andl=/contintegraldisplay (¯HׯA(x,y ))·¯andl −Az,C/contintegraldisplay (¯Hׯaz)·¯andl =/contintegraldisplay (¯HׯA(x,y ))·¯andl +Az,C/contintegraldisplay ¯H·¯aldl =/contintegraldisplay (¯HׯA(x,y ))·¯andl+Az,CJSc. (39) We see that Lex→∞ asl→∞ (arising once again from Az,C), andLexhas no meaning for an infinitely long single conductor. On the other hand, if (38) and (39) are substituted into (34), the Az,Cterms cancel and the internal inductance is expressed as Li l=/integraldisplay ¯J·¯A(x,y )dS−/contintegraldisplay/parenleftbig¯HׯA(x,y )/parenrightbig ·¯andl (40) which is finite because the the ln(l)terms are no longer present andA(x,y )is finite. Recall that since Az,Cis a constant in x andy, it does not appear explicitly in ¯H=1 µ∇× ¯A. Thus, the internal inductance of an infinitely long conductor has a unique finite value. REFERENCES [1] C. T. A. Johnk, Engineering Electromagnetic Fields and Waves .N e w York: Wiley, 1975, ch. 5. [2] H. W. Johnson and M. Graham, High-Speed Digital Design: A Handbook of Black Magic . Englewood Cliffs, NJ: Prentice-Hall, 1993, pp. 191– 192. [3] I. Catt, D. Walton, and M. Davidson, Digital Hardware Design . London, U.K.: MacMillan, 1979. [4] E. Bogatin, Signal Integrity Simplified . Upper Saddle River, NJ: Prentice-Hall, 2004. [5] S. H. Hall, G. W. Hall, and J. A. McCall, High Speed Digital System Design: A Handbook of Interconnect Theory and Design Practices .N e w York: Wiley, 2000. [6] C. R. Paul, Introduction to Electromagnetic Compatibility .N e w Y o r k : Wiley, 1992. [7] A. Scarlatti and C. L. Holloway, “An equivalent transmission line model for high speed digital transmission lines,” vol. 43, no. 4, pp. 504–514, Nov. 2001. [8] D. B. Kuznetsov and J. E. Schutt-Aine, “Optimal transient simulation of transmission lines,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. , vol. 43, no. 2, pp. 110–121, Feb. 1996. [9] D. B. Kuznetsov, “Efficient circuit simulation of nonuniform transmission lines,” IEEE Trans. Microw. Theory Tech. , vol. 46, no. 5, pp. 546–550, May 1998. HOLLOW AY AND KUESTER: DC INTERNAL INDUCTANCE FOR A CONDUCTOR 7 [10] K. M. Coperich, A. C. Cangellaris, and A. E. Ruehli, “Rigorous model- ing of the frequency dependence of ohmic losses in high-speed electri- cal interconnections,” in Proc. IEEE Int. Symp. Electromagn. Compat. , Washington, DC, Aug. 21–25, 2000, pp. 301–306. [11] F. Y . Chang, “Waveform relaxation analysis of nonuniform lossy trans- mission lines characterized with frequency-dependant parameters,” IEEE Trans. Circuits Syst. , vol. 38, no. 12, pp. 1484–1500, Dec. 1991. [12] C. S. Yen, Z. Fazarinc, and R. L. Wheeler, “Time-domain skin-effect model for transient analysis of lossy transmission lines,” Proc. IEEE , vol. 70, no. 7, pp. 750–757, Jul. 1982. [13] P. Silvester, “Model network theory of skin effect in flat conductors,” Proc. IEEE , vol. 54, no. 9, pp. 1147–1151, Sep. 1966. [14] S. Kim and D. P. Neikirk, “Compact equivalent circuit model for the skin effect,” in Proc. 1996 IEEE-MMT-S Int. Microw. Symp. , San Francisco, CA, Jun. 17–21, 1996, pp. 1815–1818. [15] S. Kim and D. P. Neikirk, “Time domain multiconductor transmission line analysis using effective internal impedance,” in Proc. IEEE 6th Topical Meeting Electr. Perform. Electron. Packag. (EPEP 1997) , San Jose, CA, Oct. 27–29, pp. 255–258. [16] H. A. Wheeler, “Formulas for the skin effect,” Proc. IRE , vol. 30, pp. 412– 424, 1942. [17] J. A. Roden, C. R. Paul, W. T. Smith, and S. D. Gedney, “Finite-difference, time-domain analysis of lossy transmission lines,” IEEE Trans. Electro- mag. Compat. , vol. 38, no. 1, pp. 15–24, Feb. 1996. [18] D. F. Williams, J. E. Rogers, and C. L. Holloway, “Multiconductor trans- mission line characterization: Representation, approximations, and accu- racy,” IEEE Trans. Microwave Theory Tech. , vol. 47, no. 4, pp. 403–409, Apr. 1999. [19] C. Gordon, T. Blazeck, and R. Mittra, “Time-domain simulation of multi- conductor transmission lines with frequency-dependent losses,” IEEE Trans. Comput.-Aided Design , vol. 11, no. 11, pp. 1372–1387, Nov. 1992. [20] Y . Eo and W. R. Eisenstadt, “High-speed VLSI interconnect modeling based on S-parameter measurements,” IEEE Trans. Compon., Hybrids, Manuf. Technol. , vol. 16, no. 5, pp. 555–562, Aug. 1993. [21] N. S. Nahman and D. R. Holt, “Transient analysis of coaxial cables using the skin effects approximation A +B√s,”IEEE Trans. Circuit Theory , vol. CT-19, no. 5, pp. 443–450, Sep. 1972. [22] D. F. Williams and C. L. Holloway, “Transmission-line parameter ap- proximation for digital simulation,” IEEE Trans. Electromagn. Compat. , vol. 43, no. 4, pp. 466–470, Nov. 2001. [23] A. R. Djordjevic, T. K. Sarkar, and R. F. R. Harrington, “Analysis of lossy transmission lines with arbitrary nonlinear terminal networks,” IEEE Trans. Microw. Theory Tech. , vol. MTT-34, no. 6, pp. 660–666, Jun. 1986. [24] R. E. Matick, Transmission Lines for Digital and Communication Net- works . New York: IEEE Press, 1969. [25] P. C. Marnusson, G. C. Alexander, and V . K. Tripathi, Transmission Lines and Wave Propagation . Boca Raton, FL: CRC, 1992. [26] C. R. Paul, Analysis of Multiconductor Transmission Lines , 2nd ed. New York: Wiley, 2007. [27] S.-H. You and E. F. Kuester, “Guaranteed passive direct lumped-element modeling of transmission lines,” IEEE Trans. Microw. Theory Tech. , vol. 53, no. 9, pp. 2826–2834, Sep. 2005. [28] P. L. Kalantarov and L. A. Tseitlin, Raschet Induktivnostie . Leningrad, Russia: Energoatomizdat, 1986, pp. 82–87 and 111. [29] T. D. Tsiboukis and E. E. Kriezis, “Calculation of inductance of conductors with various shapes of cross-section by direct methods of the functional analysis,” Il Nuovo Cimento , vol. 73 B, no. 2, pp. 177–188, Feb. 11, 1983. [30] M. Krakowski and Morawska, “Skin effect and eddy currents in a thin tape,” Archiv f ¨ur Elektrotechnik , vol. 66, pp. 95–98, 1983. [31] G. Antonini, A. Orlandi, and C. R. Paul, “Internal impedance of conductors of rectangular cross sections,” IEEE Trans. Microw. Theory Tech. , vol. 47, no. 7, pp. 979–985, Jul. 1999. [32] H. Chen and J. Fang, “Modeling of impedance of rectangular cross-section conductors,” in Proc. 2000 IEEE Conf. Electr. Perform. Electron. Packag. , Oct. 23–25, pp. 159–162. [33] M. Strutt, “Das magnetische Feld eines rechteckigen, von Gleichstrom durchflossenen Leiters,” Archiv f ¨ur Elektrotechnik , vol. 17, pp. 533–535, 1926. [34] M. Strutt, “Das magnetische Feld eines rechteckigen, von Gleichstrom durchflossenen Leiters,” Archiv f ¨ur Elektrotechnik , vol. 18, p. 282, 1927. [35] B. Hague, The Principles of Electromagnetism Applied to Electrical Ma- chines . New York: Dover, 1962, ch. XII. [36] E. Weber, Electromagnetic Theory, Mapping of Fields .N e w Y o r k : Wiley, 1950, vol. 1, ch. 4.[37] L. Lewin, Polylogarithms and Associated Functions .N e w Y o r k : N o r t h Holland, 1981. [38] P. Silvester, Modern Electromagnetic Fields . Englewood Cliffs, NJ: Prentice-Hall, 1968, p. 172. [39] H. A. Haus and J. R. Melcher, Electromagnetic Fields and Energy .E n - glewood Cliffs, NJ: Prentice-Hall, 1989. Christopher L. Holloway (S’86–M’92–SM’04) re- ceived the B.S. degree from the University of Tennessee at Chattanooga, Chatanooga, in 1986, and the M.S. and Ph.D. degrees from the University of Colorado at Boulder, Boulder, in 1988 and 1992, re- spectively, both in electrical engineering. During 1992, he was a Research Scientist with Electro Magnetic Applications, Inc., Lakewood, CO, where he was engaged in research on theoretical anal- ysis and finite-difference time-domain modeling of various electromagnetic problems. From the fall of 1992 to 1994, he was with the National Center for Atmospheric Research (NCAR), Boulder. While at NCAR, he was engaged in research on wave propa- gation modeling, signal processing studies, and radar systems design. From 1994 to 2000, he was with the Institute for Telecommunication Sciences (ITS), U.S. Department of Commerce, Boulder, where he was enaged in research on wave propagation studies. Since 2000, he has been with the National Institute of Stan- dards and Technology (NIST), Boulder, where he has been engaged in research on electromagnetic theory. He is also on the Graduate Faculty at the University of Colorado at Boulder. His current research interests include electromagnetic field theory, wave propagation, guided wave structures, remote sensing, numer- ical methods, and electromagnetic compatibilty (EMC)/electromagnetic inter- ference issues. Dr. Holloway is currently serving as Co-Chair for Commission A of the International Union of Radio Science and is an Associate Editor for the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY . He was the Chairman for the Technical Committee on Computational EMC (TC-9) of the IEEE EMC Society from 2000 to 2005, served as an IEEE Distinguished Lecturer for the EMC Compatibility Society from 2004 to 2006, and is currently serving as Co-Chair for the Technical Committee on Nano-Technology and Advanced Ma- terials (TC-11) of the IEEE EMC Society. He was the recipient of the 2008 IEEE EMC Society Richard R. Stoddart Award, the 2006 Department of Commerce Bronze Medal for his work on radio wave propagation, the 1999 Department of Commerce Silver Medal for his work on electromagnetic theory, and the 1998 Department of Commerce Bronze Medal for his work on printed circuit boards. Edward F. Kuester (S’73–M’76–SM’95–F’98) re- ceived the B.S. degree from Michigan State Univer- sity, East Lansing, in 1971, and the M.S. and Ph.D. degrees from the University of Colorado at Boulder, Boulder, in 1974 and 1976, respectively, all in elec- trical engineering. Since 1976, he has been with the Department of Electrical, Computer, and Energy Engineering at the University of Colorado at Boulder, where he is cur- rently a Professor. In 1979, he was a Summer Faculty Fellow at the Jet Propulsion Laboratory, Pasadena, CA. From 1981 to 1982, he was a Visiting Professor at the Technische Hogeschool, Delft, The Netherlands. In 1992 and 1993, he was an Invited Professor at the ´Ecole Polytechnique F ´ed´erale de Lausanne, Switzerland. He has held the position of Visiting Scientist at the National Institute of Standards and Technology (NIST), Boulder, in 2002, 2004, and 2006. He is the coauthor of one book, author of chapters in two others, and has translated two books from Russian. He is the coholder of two U.S. patents, and author or coauthor of more than 70 papers in refereed technical journals. His current research interests include the modeling of electromagnetic phenomena of guiding and radiating structures, applied mathematics, and applied physics. Dr. Kuester is a member of the Society for Industrial and Applied Mathe- matics (SIAM) and Commissions B and D of the International Union of Radio Science (URSI).