Analysis of formulas to calculate the AC inductance of different configurations of nonmagnetic circular conductors

Transcription

1 Analysis of fomulas to calculate the AC inductance of diffeent configuations of nonmagnetic cicula conductos Fancesca Capelli*, Jodi-Roge Riba** *Electical Engineeing Depatment, Univesitat Politècnica de Catalunya, Univesitat Politècnica de Catalunya, 08 Teassa, Spain. **Coesponding autho: Electical Engineeing Depatment, Univesitat Politècnica de Catalunya, Igualada, Spain, Tel.: ; fax: addess: (J.-R. Riba) ABSTRACT The inductance of single- and multi-conductos intended fo electonic devices, powe tansmission and distibution o gounding, lightning and bonding systems geatly depends on the specific geomety and the supply fequency. It is also influenced by skin and poximity effects. The inductance is an impotant design paamete, since it significantly influences the voltage dop in the conducto, thus aising eactive powe consumption and limiting the conductos cuent-caying capacity. Although thee exist some intenationally accepted appoximated and exact fomulas to calculate the AC inductance of conductos in fee ai, its accuacy and applicability has been seldom analyzed in detail in the technical liteatue, which is done in this pape. Since such fomulas can be used fo a wide divesity of conductos configuations and unde diffeent opeating conditions, it is highly desiable to evaluate thei applicability. This evaluation is caied out by compaing the esults povided by the fomulas with data povided by finite element method (FEM) simulation. The esults povided in this pape pove that FEM esults can be easily applied to a wide divesity of conductos configuations and fequencies, thus being moe geneal and often moe accuate than the esults povided by the fomulas. KEYWORDS Inductance, finite element method, eactive powe, voltage dop.. INTRODUCTION Conductos supplied fom an altenating cuent (AC) powe souce often pesent nonunifom cuent densities because of the induced eddy cuents which cause the skin and poximity effects [] [4]. Unde high-fequency supply, the electic cuent within a conducto is foced to flow within a small faction of its coss section since eddy cuents and thei associated magnetic fields induced within the conducto alte the cuent distibution acoss the conducto. It is well known that the cuent distibution within a conducto has a geat influence on its esistance and inductance [5]. As the powe supply fequency is inceased, the cuent within a conducto tends to cowd towads the oute suface [6], theefoe inceasing its effective esistance and deceasing the inne inductance. Both components of the conducto s impedance, that is, the eactance and the esistance geneate voltage dops, which ae especially impotant at high fequency opeation, thus limiting the cuent-caying capacity and inceasing the opeating tempeatue due to the active and eactive components of the electic powe consumed by the conducto itself. Inductance plays a key ole in powe cables and lines, as well as in lightning, gounding and bonding potection systems, especially at high-fequency opeation. In the technical liteatue the change of the AC inductance in conductos has been less studied than the change of the AC esistance, pobably because the inductance is highly influenced by the specific geomety of the analyzed poblem. In lightning, gounding and bounding systems, the effects of the inductance ae especially significant unde the pesence of high fequency impulse cuents, since they geneate a fast voltage dop along the conducto, thus hampeing the impulse cuent to flow towads the gounding electodes. In this case, due to the inductive effect, the gounding system is no longe equipotential [7]. This effect limits the effectiveness of the abovementioned potection systems used to avoid dangeous potential diffeences between diffeent conducto objects in an electical installation [8]. To ensue optimum pefomance, suge potection systems equie a low inductance path to the gound. It is theefoe vey impotant to have a tool to accuately detemine the inductance of the aangement analyzed. It is also well-known that the oveall impedance of a gounding electode is geatly influenced by its geomety and eath esistivity, the analysis of the impact of these paametes not being coveed by this pape. The inductance highly elies on the aea enclosed by the cuent path. In the case of an isolated conducto opeated at high fequency, the cuent is foced to flow though the oute cicumfeence of the conducto due to the skin effect [9]. Since the cuent is confined in a small faction of the conducto s total coss section, its effective o AC esistance tends to incease wheeas the effective intenal inductance tends to decease [0]. In the technical liteatue one can find both exact and simplified fomulas to calculate the inductance in conductos, some of which ae evaluated in this pape. Available analytical fomulas cannot deal with effects such as non-cicula coss-sectional

2 geometies, multi-conducto systems o unbalanced multi-phase conductos among othes. Unfotunately, due to the mathematical complexity involved, analytical fomulas ae only available fo the simplest geometies [], a limited ange of conductos dimensions and a esticted inteval of fequencies since they often assume a constant cuent density acoss the conducto s coss section []. Most of the analyzed fomulas assume that the conductos ae placed in a homogenous medium and theefoe they do not take into account othe effects such as the influence of a conducting gound plane [] o the poximity due to the pesence of neaby conductos, othewise inductance calculations would be moe complex. Depending on the distance between the conducting gound and the analyzed conductos, system inductance [4] can be substantially educed since accoding to the image theoy, the gound effect may be modeled by equivalent conductos placed undeneath the gound, though which flow a cuent in opposition to that of the souce. As a consequence, it is equied to apply numeical methods specifically intended fo solving this type of poblems, such as the finite element method (FEM) which can help in minimizing the abovementioned limitations. FEM simulations ae well suited to deal with the application consideed in this pape since they ae vey flexible and poweful and allow dealing with a wide ange of geometies and fequencies even taking into account of eddy cuent effects [5]. It is intenationally ecognized that FEM simulations povide accuate solutions in poblems involving powe conductos, tansfome windings o inductos [6], [7] whee most of the fomulas analyzed in this pape can povide inaccuate esults due to the poximity effect, especially in tansfomes and otating machines windings. This pape evaluates some of the fomulas to detemine the AC inductance of diffeent conductos shapes found in the technical liteatue and thei accuacy is compaed against esults povided by FEM simulations. This pape is a continuation of the wok pesented in [] in which the eal pat of the impedance of diffeent conductos configuations, that is the AC esistance, is analyzed in the 0 Hz MHz inteval. This fequency ange satisfies the quasi-stationay condition, so the displacement cuent can be neglected [9]. At the autho s knowledge thee ae vey few published scientific-technical woks analyzing the applicability and accuacy of such fomulas to calculate the AC inductance of diffeent conductos configuations, despite of the impotance and the pactical consequences of the inductance in conductos intended fo powe systems, thus this pape makes a contibution in this topic. It also poves that FEM simulations allow calculating the inductance in many conductos configuations taking into account both skin and poximity effects, thus offeing flexibility and accuacy, which in most configuations is not possible by means of analytical fomulas.. PARTIAL SELF-INDUCTANCE OF AN ISOLATED RECTILINEAR CONDUCTOR IN FREE AIR Inductance can be defined as the elationship between the magnetic flux though a suface bounded by a closed path and the cuent that geneates this flux. Inductance is usually associated with loops o coils although a staight segment of a conducto o a potion of a loop also has its own inductance, which is known as patial inductance. In this case it is assumed that the staight segment foms a loop whee the etun segment is at infinity. The patial inductance of the conducto segment can be defined as the atio between the magnetic flux geneated by the conducto s cuent between this segment and infinity and the conducto s cuent [8]. The concept of patial inductance allows calculating, fo example, the inductance of an infinitely long and isolated conducto [9] whee thee is no loop cuent. In this case, it is possible to calculate the expected voltage dop due to the patial inductance of the conducto. Unde AC supply, and especially at high fequency opeation o fo lage coss sections [0], the cuent density in any isolated conducto placed in fee ai tends to be concentated nea the conducto s suface due to the eddy cuents induced by the altenating magnetic field accompanying the cuent in the conducto []. This effect lessens the effective coss section, thus inceasing the AC esistance and educing its patial self-inductance. The eduction of the effective coss section of the conducto has impotant pactical effects [], including a highe voltage dop, a tempeatue incease and a decease of the conducto s ampacity. Since the cuent flowing within a conducto geneates a magnetic flux in its suoundings, inside and outside of the conducto, the patial self-inductance of an isolated single conducto has two main components, that is, the intenal and extenal inductances. The total inductance o patial self-inductance of the conducto is obtained by adding the intenal and extenal inductances []. Wheeas the intenal inductance, which epesents the smallest component [] is due to the magnetic field distibution inside the conducto, the extenal inductance is due to the total cuent within the conducto, since it involves the magnetic field distibution outside the conducto. The skin effect educes the magnetic field inside the conducto, since it is confined in a smalle volume as the fequency inceases, theefoe deceasing the intenal inductance of the conducto []. At vey high fequency opeation the total conducto s inductance is theefoe almost equal to the extenal inductance [4]. The conductos dealt with in this section ae supposed to be ectilinea with isotopic nonmagnetic mateial popeties and unifom coss sections. Isolated ectilinea conducto of cicula coss section in fee ai Since infinitely long isolated solid conductos exhibit the simplest geomety, thee ae exact fomulas to detemine thei patial self-inductance consideing the skin effect. It is well-known that the skin effect must be consideed fo lage coss sections o at high fequency opeation [6], that is, when the skin depth δ is small. The skin depth can be calculated as,

3 / f 0 () σ being the electical conductivity of the conducto s mateial, μ its elative pemeability, μ 0 the pemeability of fee space and f the supply fequency. The calculation of the patial self-inductance of an isolated infinitely long ound conducto in fee ai is easonably simple when assuming a unifom cuent distibution ove the coss section of the conducto [5]. The patial self-inductance of the wie is calculated by means of the suface integal of the magnetic flux density of the conducto s segment consideed between the cente of the conducto and infinity. The symmety of this geomety makes this calculation simple, since the magnetic field is symmetic along a cicumfeence concentic with the conducto s symmety axis. When neglecting the skin effect, that is, at vey low fequency opeation, the cuent density can be assumed unifom acoss the coss section of the conducto and the intenal patial self-inductance pe unit length (L/l u ) of an infinitely long conducto becomes a constant, 0 L i, dc [H/m] () 8 whee l u is the unity length. It is woth noting that () is a constant, and thus independent of the conducto s adius. Howeve, at high fequency opeation, that is, when consideing the skin effect, the calculation is moe complex, and the following exact fomula must be applied [6] [8], 4 be( q) be'( q) bei( q) q) Li ( Li, dc ) [H/m] () q [ be'( q)] [ q)] q d /( ) whee d c is the diamete of the conducto, c, and bei, be ae, espectively, the imaginay and eal pats of the fist kind zeo-ode Kelvin functions [9], wheeas bei and be ae thei deivatives. The seies expansion in the Kelvin functions can poduce an aithmetic oveflow of the compute capacity when q is lage, that is, fo lage conducto s coss sections o at vey high fequency opeation, when the skin depth is vey small. The extenal patial self-inductance of an infinitely long ound conducto of adius can be calculated as [0], 0 Le ln( R / ) [H/m] (4) being the adius of the conducto and R the adius of the fictitious cylinde consideed to calculate the extenal patial selfinductance, which is shown in Fig.. Intenal inductance aea R Extenal inductance aea Fig.. Regions to calculate the intenal and extenal inductances in an infinitely long ound ectilinea conducto of adius. As shown in (4), the patial self-inductance depends on the abitay adius R of the selected suface, which has no physical meaning since (4) neglects the effect of the etun cuent, so in ode to obtain the inductance of the conducto in a ealistic way the complete cicuit must be taken into account, including the etun path []. As shown in Fig., the intenal patial self-inductance can be obtained fom the magnetic enegy calculated in egion, wheeas the extenal patial self-inductance in the egion adius R can be calculated fom the stoed magnetic enegy in egion. Finally, the total patial self-inductance of the conducto can be calculated by adding the intenal and extenal inductances. It will be shown that the intenal inductance is usually much smalle than the extenal inductance [], this effect being moe manifest at high fequency opeation. At low-fequency opeation, Gigsby [0] appoximates the total patial self-inductance pe unit length of an infinitely long ound conducto as, R 0 L / l L / l L / l ln( R / ) ln( R / ) ln ln( R / GMR) [H/m] (5) tot u i u e u / e GMR being the self-geometic mean adius [9], which is calculated as, GMR = e -/4 = (6) A fomula valid when the conducto length is much lage than its adius, that is l >>, is povided in [] [5], 0 L tot / l ln( l / ) Y [H/m] (7)

4 4 whee Y = /4 fo low fequency and Y = fo high fequency applications. It shows that the incease of the conducto s adius has a limited effect on the high-fequency inductance. Accoding to Dengle [6], when taking into account complete skin effect, the patial self-inductance of a staight cicula conducto of length l can be calculated as, 0 l 8 L tot / l ln 4 l l [H/m] (8) Wu et al. [7], appoximate the patial self-inductance of staight segments of ound coss section with adius and length l by, / 0 l L tot l ln l l l 4 [H/m] (9) Isolated ectilinea tubula ound conducto in fee ai Due of its geomety, a tubula ound conducto allows minimizing weight and mateial costs compaed to solid conductos, especially fo lage coss sections o at high fequency opeation. By using tubula ound conductos, the voltage dop due to the impedance of the conductos, powe loss and eactive powe can be significantly educed. Fig. shows the coss section of a ound tubula conducto. Fig.. Coss-section of a ound tubula conducto Thee exists an exact solution fo calculating the pe unit length intenal patial self-inductance of an isolated infinitely long tubula conducto, which is based on the (be, bei) and (ke, kei) functions, that is, the zeo-ode Kelvin functions of fist kind and second kind, espectively and thei deivatives (be,bei ) and (ke, kei ). When assuming that the cuent etuns outside the conducto emotely it esults in [6], [], l j m be(m ) j bei(m ) C ke(m ) jc kei(m ) imag be'(m ) j m ) C ke'(m ) jc kei'(m ) [H/m] (0) f Li, ac u be' (m ) j bei (m ) whee C ' /, m ( f ), and ke'(m ) jkei'(m ) and ae, espectively, the inside and outside adius of the tubula conducto. The DC intenal patial self-inductance of the infinitely long tubula conducto with extenal etun can be expessed as [], [6], L i, dc / l u 4 ln( / ) [H/m] () ( ) 4( ) Similaly as in (4), the extenal inductance of an infinitely long tubula ound conducto pe unit length with oute adius can be calculated as [0], 0 Le ln( R / ) [H/m] () Fo infinitely long tubula conductos, Sapongin and Pokopenko [8] poposed the following fomula to calculate the total patial self-inductance pe unit length, L ( / ) ( / ) ln( / ) tot 0 ln( R / ) [H/m] () l 4 ( / ) u In the case of infinitely long thin-wall tubula conductos () becomes [8], 0 4 Ltot t t / 6 t / 0 ln( R / ) [H/m] (4) 4 whee t = ( )/ <<.

5 5. INDUCTANCE OF A RETURN CIRCUIT OF RECTILINEAR CONDUCTORS IN FREE AIR As explained, when dealing with staight conductos, the effect of the etun path must be consideed, and then the dependence of R (adius of the extenal cylinde taken into account) in fomulas such as (4), (5) and () disappeas. In addition, the poximity effect aises, thus making moe complex the deivation of analytical fomulas.. Retun cicuit of ectilinea ound conductos Fig. shows the geomety of a etun cicuit composed of two staight conductos of cicula coss section, that is, with the cuents flowing in opposite diections and with axial distance D. I I D l Fig.. Retun cicuit fomed by two identical conductos of adius with the cuents flowing in opposite diections. The extenal inductance is associated to the magnetic flux suounding the conducto, which stoes magnetic enegy, wheeas the intenal inductance neglects the extenal flux. The extenal patial inductance pe unit length of a single infinitely long conducto unde the supposition D >> is given by [9], 0 D Le c ln [H/m] (5), D being the distance between conductos axes and the conducto s adius. Since in a etun cicuit thee ae two infinitely long conductos with cuents flowing in opposite diections, the total patial extenal inductance pe unit length is twice (5), that is, 0 D Le,c ln [H/m] (6) The total patial inductance of two infinitely long paallel conductos with opposite cuents takes into account both the intenal inductance L i,c /l u given by () and the extenal inductance L e,c /l u given by (6), thus esulting in, 8 be ( q) be '( q) bei( q) q) 0 D L / l L / l L / l ( L / l ) ln tot c u i c u e c u i dc u [H/m] (7),,,, q [ be '( q)] [ q)] Since (7) is based on the exact solution of an infinitely long isolated conducto, it doesn t take into account the poximity effect, and theefoe it is not an exact fomula. It is woth noting that (7) is not found in the technical liteatue although it is staightfowadly obtained fom () and (6). Accoding to Sakis [], when neglecting the skin effect, the total patial inductance of the infinitely long single-phase conducto esults in, D D L / l L / l L / l ln ln tot c u i c u e c u [H/m] (8),,, 4 GMR whee GMR = e -/4 = Gigsby and othe authos [0] expess the total inductance of a etun cicuit of infinitely long ound conductos as, 0 D Ltot,c ln [H/m] (9) GMR When dealing with two o moe conductos, thee is an inteaction among the magnetic fields of each conducto, thus appeaing the concept of patial mutual inductance between two conductos, which is defined as the atio between the flux linkage fom one conducto to infinity to the cuent flowing in the othe conducto. When consideing a etun cicuit in which the cuents in the two staight conductos flow in opposite diections, the patial self-inductance of the etun cicuit can be obtained fom the patial self-inductances of the individual conductos and the patial mutual inductance as [4], [40], Accoding to Rosa [40], the total inductance of the etun cicuit of ound conductos and finite length l is given by, Lcicuit Lconducto, Lconducto, M (0) M being the patial mutual inductance. If the conductos ae identical, that is, with same lengths and adii, (0) esults in, L L M () tot, c c whee L c is the patial self-inductance of each single conducto, which is calculated as in (7). The patial mutual inductance M can be expessed as,

6 6 0 l l D l D D 0 l M / l ln [ln D / l] [H/m] () D l l D The appoximation applied in () is valid when l >> D. By substituting (7) and () in () it esults in, 0 D Ltot, c / l [ln / 4 D / l] [H/m] (). Isolated tubula ound conducto with intenal etun in fee ai In this section a tubula conducto with intenal etun is analyzed. Fig. 4 shows this geomety, with inne adius of the inne and oute conductos, and, espectively. Fig. 4. Coss-section of a tubula conducto with inne etun Accoding to Russell [4], when assuming that the cuent etuns concentically inside the conducto, the intenal selfinductance pe unit length is calculated as, m m be' ( m ) L / l [ be( m ) be' ( m ) bei( m ) bei' ( m )] [ M be( m ) N bei( m ) O ke( m ) P kei( m )] i,c u 4 m ) 4 m ) (4) m bei' ( m ) [ M bei( m ) N be( m ) O kei( m ) P ke( m )] 4 m ) whee, / m ( f ) 0, S( mx) be '( mx) ke'( mx) mx) kei'( mx), T( mx) mx) ke'( mx) be '( mx) kei'( mx), mx) be ' ( mx) bei' ( mx), O S( m ) PT ( m ) M m ) OT ( m ) P S( m ) N m ) ( / ) m) [ m) S( m) m) S( m)] [ bei' ( m) m) be'( m) be' ( m O )] [ m ( ( ( [ ( ( ( ( ) S m) Y m) S m)] Y m) T m) Y m) T m)] ( / ) m) [ m) T( m) m) T( m)] [ be' ( m) m) m) be'( m )] [ m ( ( ( [ ( ( ( ( ) S m) Y m) S m)] Y m) T m) Y m) T m)] ( / ) m ) [ m ) T( m ) m ) T( m )] [ be'( m ) be' ( m ) bei' ( m ) m P )] [ m ( ) S( m ) m ) S( m )] [ m ) T( m ) m ) T m )] ( / ) m ) [ m ) S( m ) m ) S( m )] [ be' ( m ) m ) bei' ( m ) be'( m )] [ m ( ( ( [ ( ( ( ( ) S m ) Y m ) S m )] Y m ) T m ) Y m ) T m )] Although (4) consides the skin effect, it ignoes the poximity effect between the conductos. The DC pe unit length intenal inductance of the tubula conducto with intenal etun can be expessed as [6], 4 0 Li,c. dc/ lu ( ) ln( / ) 4( ) [H/m] (5) The pe unit length extenal inductance of an infinitely long ound coaxial conducto can be calculated as [4], 0 L e,c / l u ln [H/m] (6) Finally, when consideing the skin effect, the total self-inductance can be calculated by adding (4) and (6). When ignoing the skin effect, the total self-inductance pe unit length can be calculated by adding the inne self-inductance of the ound conducto which is given by (), and the extenal self-inductance given by (6), thus esulting,

7 7 0 Ltot,c ln 4 [H/m] (7) 4. RESULTS This section compaes the esults povided by the fomulas developed in Sections II and III with those povided by twodimensional FEM simulations. Despite the impotant pactical consequences of the inductance in conducto systems, thee ae vitually no published expeimental data, theefoe fo compaison puposes FEM is taken as the efeence method due to the accuate esults povided []. To this end, the two-dimensional FEMM package [4] has been used since it is easy-to-use and povides fast and accuate solutions. Futhe details about the mathematical backgound of the FEMM fomulation can be found in []. All configuations studied in this section studied ae analyzed by means of D-FEM models imposing a cicula ai peimete with a Diichlet bounday condition that foces the vecto potential A to be zeo at the oute bounday. FEM-based simulations equie discetizing the analyzed geometic domains, so they ae divided into small elements. The size of such elements depends on the dimensions of the studied conductos and the analyzed fequency. Solutions povided by the FEM simulations poposed in this wok have low computational equiements, thus poviding fast esults. The inductance can be obtained fom FEM simulations since in the case of linea magnetic egions, FEM calculates the stoed magnetic enegy as the volume integal [5], W mag B Hdv (8) v When dealing with an isolated ectilinea conducto, its patial inductance can be calculated fom the stoed magnetic enegy as, i W di L i mag L i (9) 0 Theefoe fom (8) and (9) the conducto s patial self-inductance can be calculated as, L W / i [H] (0) The analyzed D-FEM domains involving staight conductos have a depth of one mete, so by applying (0) the inductance pe unit length is diectly calculated. Theefoe it is staightfowad to genealize the esults to othe conductos lengths. 4. AC inductance of an isolated solid ound conducto in fee ai Many applications such as gounding systems, electonics, communication devices and powe systems among othes use ound solid conductos [], so it is of geat inteest to detemine the change of the inductance as a function of the fequency. Table I shows the evolution of the patial self-inductance pedicted by FEM simulations and analytical fomulas in the ange 0 Hz MHz fo a coppe ound conducto with diamete 9.8 mm. Results fom Table I show that the accuacy of (7), (8) and (9) geatly depends on the conducto length consideed, so that the accuacy inceases with the length of the conducto since fo lage conducto lengths the inductance pe unit length povided by such fomulas is vey close to the exact one, with eos in the ange.-.%. Anothe souce of eo is the skin effect. Table I clealy shows that (5), (7), (8) and (9) do not conside the skin effect because the esults they povide ae independent of the fequency although this effect is much less than that of the finite length of the conducto. TABLE I TOTAL PARTIAL SELF-INDUCTANCES PER UNIT LENGTH OBTAINED FROM EXACT AND APPROXIMATED ANALYTICAL FORMULAS AND FEM SIMULATIONS OF A COPPER ROUND SOLID CONDUCTOR WITH DIAMETER 9.8 MM ( MM ). Total patial self-inductance pe unit length L tot/l (x0-6 H/m) Fequency Exact (Hz) () + (4) FEM (5), (7) (8) (9) l = m l = m Oveflow MAPE - 0.%.% 7.-.% % Assume infinitely long conductos Neglect the skin effect The pecentage eo among the diffeent calculation systems has been calculated by means of the mean absolute pecentage mag

8 8 eo o MAPE. Note that in this case, the MAPE has been calculated with espect to the esults povided by the exact fomula as, n being the numbe of samples o fequencies calculated, and y i and fequency component and the foecast value. 4. AC inductance of an isolated tubula ound conducto in fee ai n yi yˆ i M () n y i i ŷ being espectively, the efeence value of i-th i Tubula ound conductos ae vey applied in induction funaces, outdoo substation busbas, paticle acceleatos o MRI devices among othes. Table II shows the evolution of the patial self-inductance pedicted by FEM simulations and analytical fomulas in the ange 0 Hz MHz fo a coppe tubula ound conducto with inne and oute adius 5 mm and 9.64 mm, espectively. TABLE II TOTAL PARTIAL SELF-INDUCTANCE PER UNIT LENGTH FROM EXACT AND APPROXIMATED ANALYTICAL FORMULAS AND FEM SIMULATIONS OF A COPPER ROUND TUBULAR CONDUCTOR WHEN THE CURRENT RETURNS OUTSIDE THE CONDUCTOR REMOTELY WITH R = 5 MM AND R = 9.64 MM. Total patial self-inductance pe unit length L tot/l (x0-6 H/m) Fequency (Hz) (0) + () FEM (), (4), MAPE - 0.%.5%.6% Assume infinitely long conductos Neglect the skin effect As shown in Table II, FEM esults ae moe accuate than those obtained with analytical fomulas since they also have into account the change in inductance due to the skin effect. It should be highlighted that esults povided by () and (4) ae independent of the conducto length. 4. AC inductance of a etun cicuit of cicula coss section in fee ai Retun cicuits composed of two conductos of cicula coss section ae of geat inteest since they have a wide ange of applications. Tables III and IV show the change of the total patial inductance pedicted by FEM simulations and analytical fomulas in the ange 0 Hz MHz fo a etun cicuit of cicula coss section of 00 mm spaced axially 00 mm and 500 mm, espectively. TABLE III TOTAL PARTIAL INDUCTANCE PER UNIT LENGTH FROM APPROXIMATED ANALYTICAL FORMULAS AND FEM SIMULATIONS OF A RETURN CIRCUIT COMPOSED OF TWO ROUND CONDUCTORS (D = 00 MM, R = 5.64 MM, A = 00 MM ). Total patial inductance pe unit length L tot/l (x0-6 H/m) Fequency FEM (Hz) (7), (8), (9), () l = m MAPE - 0.%.%.9% % Assume infinitely long conductos Neglect the skin effect Neglect the poximity effect TABLE IV TOTAL PARTIAL INDUCTANCE PER UNIT LENGTH FROM APPROXIMATED ANALYTICAL FORMULAS AND FEM SIMULATIONS OF A RETURN CIRCUIT COMPOSED OF TWO ROUND CONDUCTORS (D = 500 MM, R = 5.64 MM, A = 00 MM). Total patial inductance pe unit length

9 9 L tot/l (x0-6 H/m) Fequency FEM (7), (8), (9), () (Hz) l = m MAPE - < 0.%.0%.9%.5-.4% Assume infinitely long conductos Neglect the skin and poximity effects Neglect the poximity effect As shown in Tables III and IV, when the axial distance between conductos inceases, the MAPE also deceases because the poximity effect is less ponounced. It is woth noting that wheeas the esults fom () ae almost independent of the conducto length, esults fom (7), (8) and (9) ae completely independent of such vaiable. Anothe souce of eo is due to the eddy cuents effect. As summaized in Tables III and IV, the esults povided by (8), (9) and () ae independent of the fequency since they do not conside the eddy cuents effect. 4.4 Inductance of a etun cicuit of a tubula conducto with intenal etun in fee ai Coaxial conductos ae widely used in many applications including communication systems, television o ada among othes. Tables V and VI show the change of the total patial inductance pedicted by FEM simulations and analytical fomulas in the ange 0 Hz MHz fo a etun cicuit of a tubula conducto with intenal etun fo two configuations. The configuation analyzed in Table V consists of a coppe coaxial conducto with inne etun and = 5 mm, = 0 mm and = mm (see Fig. 4), wheeas the configuation analyzed in Table VI coesponds to a coppe coaxial conducto with inne etun and = 0 mm, = 5 mm and = 7 mm. These two specific configuations ae analyzed to detemine the influence of the extenal inductance tem, which depends on the / atio. TABLE V TOTAL PARTIAL INDUCTANCE PER UNIT LENGTH OBTAINED FROM APPROXIMATED ANALYTICAL FORMULAS AND FEM SIMULATIONS OF A COPPER COAXIAL CONDUCTOR WITH INNER RETURN AND R = 5 MM, R = 0 MM AND R = MM (R /R = ). Total patial inductance pe unit length L tot/l (x0-7 H/m) Fequency (Hz) FEM (4) + (6), (5) + (6), (7), Oveflow MAPE -.57%.5%.8% Assume infinitely long conductos Neglect the skin and poximity effects Neglect the poximity effect TABLE VI TOTAL PARTIAL INDUCTANCE PER UNIT LENGTH OBTAINED FROM APPROXIMATED ANALYTICAL FORMULAS AND FEM SIMULATIONS OF A COPPER COAXIAL CONDUCTOR WITH INNER RETURN AND R = 0 MM AND R = 5 MM AND R = 7 MM (R /R =.5). Total patial inductance pe unit length L tot/l (x0-7 H/m) Fequency (Hz) FEM (4) + (6), (5) + (6), (7),

10 Oveflow.8. MAPE -.% 9.%.6% Assume infinitely long conductos Neglect the skin and poximity effects Neglect the poximity effect Results fom Tables V and VI clealy show that the accuacy of (4) + (6), (5) + (6) and (7) geatly depends on the specific configuation. The main contibution in the total inductance is due to the extenal inductance tem given by (6), which is highly influenced by the atio /. Theefoe the configuation analyzed in Table V shows a highe inductance than that in Table VI because the fist configuation has a highe atio /. As deduced fom Tables V and VI, fo fequencies 0 khz and above the poximity effect becomes moe impotant and thus the diffeence between FEM esults and those attained by applying (4) + (6) is not negligible. The D meshes of the conductos configuations analyzed in this section though FEM simulations consist of a vaiable numbe of D tiangula element, which depends on the geomety and dimensions of the poblem and the fequency analyzed. The meshes of the geometies analyzed in Tables V and VI consist of a vaiable numbe of tiangula elements within (0 Hz) and ( MHz). The computational time equied to pefom such simulations was between 0 s and 90 s when using a compute with an Intel Xeon CPU, E5-66 pocessos and GB of RAM memoy. 5. CONCLUSION This pape has analyzed the accuacy of diffeent exact and appoximate fomulas to calculate the inductance of diffeent conductos configuations caying AC cuents in a boad fequency inteval. The esults povided by these fomulas have been compaed with those obtained though two-dimensional FEM simulations, which ae used as the efeence method due to its accuacy and flexibility. It has been shown that analytical exact fomulas ae only available fo the simplest and symmetical geometies, and that when this symmety is lost o when poximity effects aise, the accuacy of such fomulas is highly affected. Results attained have shown the limitations of the commonly-used analyzed fomulas in the fequency ange compising 0 Hz and MHz, and have shown which specific configuations ae moe suitable to minimize the total inductance. ACKNOWLEDGMENTS The authos would like to thank the Spanish Ministy of Economy and Competitiveness and Genealitat de Catalunya fo the financial suppot eceived unde poject RTC and Doctoat Industial unde poject DI-04-0, espectively. REFERENCES [] W. I. Middleton and E. W. Davis, Skin effect in lage standed conductos at low fequencies, J. Am. Inst. Elect. Eng., vol. 40, no. 9, pp , Sep. 9. [] V. T. Mogan, The Cuent Distibution, Resistance and Intenal Inductance of Linea Powe System Conductos A Review of Explicit Equations, IEEE Tans. Powe Deliv., vol. 8, no., pp. 5 6, Jul. 0. [] J.-R. Riba, Analysis of fomulas to calculate the AC esistance of diffeent conductos configuations, Elect. Powe Syst. Res., vol. 7, pp. 9 00, Oct. 05. [4] J.-R. Riba, Calculation of the ac to dc esistance atio of conductive nonmagnetic staight conductos by applying FEM simulations, Eu. J. Phys., vol. 6, no. July, pp. 0, 05. [5] L. Cohen, The influence of fequency on the esistance and inductance of solenoidal coils, Dolezalek, Ann. de Phys., vol., pp. 6 78, 90. [6] A. W. Ba, Calculation of Fequency-Dependent Impedance fo Conductos of Rectangula Coss Section, AMP J. Technol., vol., pp. 9 00, 99. [7] J. He, R. Zeng, and B. Zhang, Methodology and Technology fo Powe System Gounding - Jinliang He, Rong Zeng, Bo Zhang. Singapoe: John Wiley & Sons Singapoe Pte. Ltd., 0. [8] Eitech, Gounding & Bonding Fo Electical Systems. p. 70, 05. [9] O. Coufal, Cuent density in two solid paallel conductos and thei impedance, Elect. Eng., vol. 96, no., pp , Sep. 04. [0] E. B. Joffe and K.-S. Lock, Gounds fo Gounding: A Cicuit-to-System Handbook. Hoboken, New Jesey: John Wiley & Sons, Inc, 00. [] P. Patel, Calculation of Total Inductance of a Staight Conducto of Finite Length, Phys. Educ., vol. July Sep, pp. 9 98, 009. [] O. Coufal, On Resistance and Inductance of Solid Conductos, J. Eng., vol. 0, pp. 4, 0. [] X. Jia, L. Liu, and G. Fang, The Finite-Conducting Gound s Effect on the Inductance of a Rectangula Loop, J. Sensos, vol. 06, pp., 06. [4] J.-L. Schanen, C. Guein, J. Roudet, and G. Meunie, Influence of a conductive plane on loop inductance, IEEE Tans. Magn., vol., no., pp. 7 0, May 995. [5] R. Sita, Ž. Štih, and Z. Valković, Expeimental veification of quasi-nonlinea modeling of magnetic steel fo time-hamonic eddy cuent loss calculation, Elect. Eng., pp. 7, May 06. [6] U. Patel and P. Tiveio, Accuate Impedance Calculation fo Undegound and Submaine Powe Cables using MoM-SO and a Multilaye Gound Model, IEEE Tans. Powe Deliv., pp., 05.

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