Calculation of per-unit-length resistance and internal inductance in 2-D skin-effect current driven problems

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 44, NO. 4, NOVEMBER 2002 529 Calculation of Per-Unit-Length Resistance and Internal Inductance in 2-D Skin-Effect Current Driven Problems Massimo Vitelli Abstract—The study of two-dimensional (2-D) skin-effect transient problems by means of a formulation based on a magnetic and on a scalar potential is presented. It is vector potential shown that such 2-D problems in the cross-sectional plane of the conductors cannot be treated as voltage driven but only as current driven. As an application of the proposed method, the equivalent per-unit-length resistance and internal inductance of conductors carrying typical switch-mode power supplies nonsinusoidal current waveforms have been evaluated. The analysis clearly reveals that they depend both on switching frequency and duty cycle, and are rather different from dc and ac sinusoidal equivalent parameters. Time-varying instantaneous ohmic resistances and internal inductances can be also defined in sinusoidal steady-state as well as in generic operating conditions. A Index Terms—Inductance, resistance, skin effect, switch-mode power supplies (SMPS). I. INTRODUCTION T HE evaluation of the exact value of the resistance as well as that of the internal inductance characterizing conductors working in different operating conditions, as concerns the parameters describing the shape of the currents flowing in them, is of great interest in the design of many practical devices such as, for example, induction machines and power electronic systems. In particular, in dc–dc switching converters, the increasing high-frequency spectral content characterizing soft-switching techniques, which are widely adopted in order to obtain converters characterized by lower losses and higher switching frequencies, causes the enhancement of the impedance of connecting wires which cannot be neglected any more. This is due to the nonuniform current distribution (skin effect), over the conductor cross section, occurring at those frequencies where the cross-sectional dimensions are much larger than a skin depth where is the electric conductivity and is the magnetic permeability of the conductor. Moreover, a lumpedparameter circuit model is no more adequate to describe such connecting conductors and a proper multiconductor transmission-line (MTL) model must be used [1]. In particular the perunit-length parameters of inductance, capacitance, conductance and resistance, that distinguish one MTL from another, must be properly evaluated. This paper is focused on the determina- Manuscript received December 7, 2001; revised May 27, 2002. The author is with the Seconda Università di Napoli, DII, Naples 81031 Aversa (CE), Italy (e-mail: vitelli@unina.it). Digital Object Identifier 10.1109/TEMC.2002.804771 tion of the per-unit-length resistance and internal inductance. It is well known that lossy conductors implicitly invalidate the basic transverse electromagnetic (TEM) field-structure assumption on which the MTL model relies. However, if the conductor losses are small, the resulting field structure can be considered almost TEM (quasi-TEM assumption) and the MTL model reasonably adequate [1]. In a typical skin effect three-dimensional (3-D) problem, a time-varying current flows in one or more conductors with the current density distribution unknown, and either the voltage over the conductors (voltage driven problems) or the overall current through them (current driven problems) known. Different 3-D formulations have been proposed in the literature in order to study these problems [2]–[4]. Nevertheless, in transmission-line problems, the conductors are assumed infinitely long and the computation of the per-unit-length parameters becomes an easier two-dimensional (2-D) problem in the transverse plane of the line. This is the fundamental motivation why, in the following, it will be considered a 2-D formulation. Even if the calculation of per-unit-length parameters for an isolated conductor at sinusoidal steady-state operation seems to be a well-known result [5]–[8], in this paper, some aspects will be stressed which, to the best of the author’s knowledge, have not yet been discussed in the literature. II. FORMULATION Consider a single infinite (along ), homogeneous and linear conductor in the – plane with permeability and conductivity . Such conductor is surrounded by an unbounded of permeability . Since we connonconducting domain directed along the -axis, we can assider a total current sume that the flux density , the electric field , and the current density are such that in (1.a) in in (1.b) (1.c) that is, and are essentially -directed while lies in the plane. The approach adopted in this paper is based on the use of a magnetic vector potential and a scalar potential [5]; is related in the usual way to 0018-9375/02$17.00 © 2002 IEEE in (2.a) 530 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 44, NO. 4, NOVEMBER 2002 and, from Faraday’s law, can be expressed in term of in It is worth noting that, since -components and (2.b) is -directed, has no in and (3) can have a From (2b), it can be easily deduced that also must nonzero component only along the axis; moreover, be independent on . This means that the scalar potential function can depend on and in the following way only: in (4) Fig. 1. Mapping transformation. nates does not affect the values of the scalar function used for the formulation of the problem. What is known, in current driven problems, is the total current From (4) we get (9) in (5) The magneto-quasi-static (MQS) approximation of Maxwell equations [5] leads to from which we get in in (10) (6) which, by using (5), and after a projection on the -axis, can be rewritten as in in (7) must be continuous across the boundary of Of course, must vanish at infinity. To study the the conductor; moreover above unbounded scalar problem by means of the finite element method (FEM), a technique based on the use of a suitable spatial transformation has been adopted, which converts the initial open boundary problem into a closed boundary one and is characterized by very good accuracy of results. Moreover, such a technique is quite easy to implement in a FEM-based numerical code [9]. The whole space is subdivided into two subdomains: and . is a circle of finite radius and contains and part of while extends to infinity and contains air only. The adopted transformation is defined by the following relations: is the cross section of the conductor carrying . where Two considerations should be stressed at this point, as follows. , by 1) The system (7) could be numerically solved in is a known means of a standard FEM formulation, if term. Unfortunately, is not known unless the contribution of the integral term appearing in the right-hand side of (10) vanishes, as it happens for example, at the sinusoidal steady-state at very low frequencies as shown in Section III-B. 2) 2-D voltage driven problems are not physically sound since, due to (1a), a unique voltage along the -axis could not be assigned but, on the contrary, it should be a function of both the - and -coordinates. By substituting (10) in (7) we get in in (8) and the correThe points belonging to the circle of radius sponding image points, obtained by using transformation (8), is mapped onto a finite subdomain are coincident; instead which is a ring of radii and as shown in Fig. 1. can be chosen arbitrarily and is not required to be very large [9]; on the contrary, computational economy dictates a boundary of just large enough to contain the entire device to be analyzed. Of course, on the boundary of homogeneous Dirichlet conditions have to be adopted since the transformation of coordi- (11) in which the right-hand side is obviously known. The presence of the integral term in the left-hand side of (11) gives rise to a FEM formulation characterized by a nonsparse mass matrix obviously leading to additional, undesirable memory storage requirements and consequently to time-expensive numerical simulations [10]. If one is interested in sinusoidal steady-state analysis only, the easier standard parabolic equation (7) can be used as explained , the in the following. Given the impressed sinusoidal current following system of partial differential equations can be solved: in in (12) VITELLI: CALCULATION OF PER-UNIT-LENGTH RESISTANCE AND INTERNAL INDUCTANCE 531 TABLE I SHAPE AND PARAMETERS OF CURRENT WAVEFORMS ADOPTED FOR SIMULATIONS where . It is worth noting that doesn’t coincide with which is not known a priori; is just a sinusoidal forcing term in (12) proportional to . , As a consequence, the obtained fields distribution , , , and are different from the corresponding ones of the original problem. Also, the total current is different from the real . However, since we are interested in the impressed current guaransinusoidal steady-state solution only, the chosen tees, due to the linearity of the considered problem, that the of will be sinusoidal steady-state current component . To obtain the real steady-state at the same frequency of , it is sufficient to spatial distributions of the fields due to , , , and (the sinusoidal multiply the fields steady-state components of , , , , and , respec(where and are the rms tively), by the scalar and , respectively), and to shift their phase values of where and are the instantaneous angle by a quantity phase angles of and , respectively. Of course, the sinusoidal steady-state analysis could be carried out much more efficiently in the frequency domain rather than in the time domain by using one of the following models: in in (13) in in (14) where dotted (complex) quantities represent the phasors of the corresponding time-domain variables. Equations (13) and (14) represent the phasor forms of (7) and (11), respectively. The FEM solution of (14) is obviously characterized, as well as the FEM solution of (11), by strong memory-storage requirements. Therefore, the adoption of formulation (13) instead of (14) is preferable since it requires only some trivial scaling of fields amplitudes and shifting of their phase angles, as explained for (7). Nevertheless, in Section III-B, in order to obtain the sinusoidal steady-state solution, time-domain formulation (7) has been used instead of frequency-domain formulation (13), in order to stress some aspects related to the instantaneous field patterns of , at sinusoidal steady-state, which are essential for a deeper comprehension of skin-effect problems. is a distorted As will be discussed in Section III-A, when nonsinusoidal current waveform, the direct solution of (11) in the time domain is much more efficient than the frequency-domain approach of (13), coupled with the Fourier series expan. Moreover, it is worth noting that, in the case of sion of nonperiodic current waveforms , the frequency-domain apis only theproach coupled with the Fourier transform of oretically but not practically possible. This is the reason why, in Section III-A, the calculation of per-unit-length parameters in the presence of typical switch-mode power supplies (SMPS) current waveforms has been carried out by using the time-domain approach of (11). III. RESULTS AND DISCUSSION A. Transient Analysis Modern SMPS, based both on hard- and soft-switching converters, generate strongly distorted current waveforms with increasing frequency spectral content. In addition to the intrinsic distortion of triangular, trapezoidal and quasisquare waveforms, the spectral content of currents of SMPS increases because of power factor correction (PFC) and duty-cycle modulation techniques adopted for electromagnetic interference (EMI) reduction. What is of interest from the design point of view is the knowledge of global effects produced by the distortion on scalar equivalent R–L–C parameters of passive and active components. In general, all the equivalent R–L–C parameters, the principal ones as well as the parasitic ones, depend on which voltage/current waveforms the devices are subjected to. Among the parameters of greatest interest for the design of SMPS, the resistance 532 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 44, NO. 4, NOVEMBER 2002 (a) (a) (b) Fig. 2. (a) Equivalent resistance r versus current waveform type. (b) Internal inductance l versus current waveform type. Values marked by circles have been obtained in distorted steady-state operating conditions by assuming that the waveforms are periodic. Values marked by squares have been obtained by assuming that the current shapes are not replicated with period T , that is, they have been evaluated just in the first period, during transient operation. Fig. 3. AC. resistance r ; solid line—theoretical predictions; dashed line—numerical simulations. plays a primary role because of its reflections on the efficiency, especially if a high-efficiency target tightly restricts the total losses budget to be shared among RI distribution losses, diodes and MOSFET’s losses, wounded components losses, equivalent series resistance (ESR) losses, etc. For this reason, the accurate (b) Fig. 4. (a) i(t) (f = 100 Hz). (b) i (t) (f = 100 Hz). calculation of the equivalent resistance of conductors in distorted operating conditions (DOC) is necessary. Ohmic losses in conductors under dynamic conditions are mainly ruled by the skin effect. Theoretically, the skin effect in DOC can be studied in the time domain as well as in the frequency domain. In fact, (characterized by a pea given periodic impressed current riod ) can be expanded in a Fourier series: ; since the problem under study corresponding to can is linear, the distribution of be obtained as: where is the distribution of current density correof . The persponding to the harmonic component at frequency is inunit-length ohmic loss dependent of (or orthogonal to) the loss at other frequencies; can this means that the loss at each harmonic frequency as be calculated individually to get the total loss where is the per-unit-length resistance at frequency and is the rms value of . Therefore, the equivalent resistance can be found as . Since the amplitude of harmonic components decreases rapidly with the harmonic number for most waveforms, it is often thought that “only a few” harmonics need to be calculated to obtain reasonable accuracy. This is not true in the case of harmonic rich waveforms such as triangle, pulse, and VITELLI: CALCULATION OF PER-UNIT-LENGTH RESISTANCE AND INTERNAL INDUCTANCE (a) (b) Fig. 5. (a) i(t) (f = 10 kHz). (b) i (t) (f = 10 kHz). square waves characterized by very fast rise and/or fall times; in addition it should be considered that, while decreases, instead increases with . Moreover, although the individual of harmonics of higher and higher order contribution to is vanishingly small, the cumulative effect of the large numbers of harmonics required to reproduce fast rise and fall times, characterizing current waveforms of present-day switching converters, is not trivial. As a consequence of the above considerations, some hundreds or even thousands of harmonics need to be taken into account in the case of current waveforms characterized by very fast rise and/or fall times to get accurate numerical results [11]. Therefore, a frequency-domain approach would require the numerical solution of the field problem as many times as the unpredictable number of harmonics required . In addition, also the accurate to get an accurate value of numerical evaluation of the Fourier series of the periodic imis required; in fact, the analytical determipressed current nation of the Fourier series expansion of is affordable only for very simple waveforms but not for example in the case of , , and of Table I referPFC current waveforms like ring to three different operation modes of PFC regulators, continuous conduction mode, boundary mode, and discontinuous conduction mode, respectively [12]. On the basis of these con- 533 (a) (b) Fig. 6. (a) i(t) (f = 1 MHz). (b) i (t) (f = 1 MHz). siderations it can be stated that, in the case of current waveforms characterized by very fast rise and/or fall times as it happens in present-day switching converters, the time-domain approach is much more efficient than the frequency-domain approach which is highly impractical. Moreover, in the case of nonperiodic current waveforms such as those ones characterizing the operation of chaotic switching converters [13], [14] or those ones typical of switching converters working with strongly time-varying loads (such as computer power supplies), the frequency-domain approach is only theoretically but not practically possible. As shown in the following examples, the losses into the conductor and consequently its equivalent resistance depend on all the elements characterizing the shape of the current itself, namely its geometric pattern (triangular, trapezoidal, squared), the switching frequency, and the duty cycle. The examples of application presented concern the calculation of the equivalent resistance for various distorted current shapes taken from dc-dc switching converters. In particular, a 6.45 10 mm; conducconductor AWG 16 (radius 5.8 10 S/m) is considered. For the sake of comtivity pleteness also, the numerical results concerning the equivalent internal inductance will be presented even if greater emphasis is 534 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 44, NO. 4, NOVEMBER 2002 (a) (a) (b) (b) Fig. 8. (a) f = 100 Hz. (b) f = 1 MHz. Fig. 7. (a) i(t) (f = 10 MHz). (b) i (t) (f = 10 MHz). given to the discussion of features concerning conductor resisand inductance tance. Per-unit-length equivalent resistance are defined as (15.a) (15.b) . The analwhere is the rms value of the injected current ysis has been carried out by using 19 different current wave; the shapes and the forms, each identified by a label parameters of such currents are reported in Table I. are typical Waveforms with periodic shapes in dc-dc converters. The duty ratio indicates the interval , normalized with respect to the period during which ; the peak value has been normalized to 1 for all the shapes. The results of computations, carried out by means of a FEM formulation of (11), are shown in Fig. 2(a) and (b). and are plotted versus the waveform The values of identification number (ID.). The values assumed by in distorted steady-state operating conditions are marked by circles in Fig. 2(a) and (b); they have been obtained in the is following way: a given current waveform and the time-domain solution is allowed turned on at to progress until the steady state is reached, whereupon the (inductance ) values are calculated. equivalent resistance The theoretical value of equivalent resistance (equivalent internal inductance ) for sinusoidal currents – , obtained by using the closed-form expression given in [5], has been also reported in Fig. 2(a) and (b), marked with the symbol “ ”. Fig. 2(a) and (b) shows that, in case of sinusoidal waveforms, and yielded by computations are in good the values of agreement with theoretical predictions: indeed, the shapes – have been considered also to test the numerical method using a known valid reference solution. The comparison between sinusoidal operating conditions (SOC) and DOC values made in Fig. 2(a) puts in evidence that is rather of and exhibits an evident dependence on the different from duty ratio . The decrease of versus is systematic and can be justified by considering that a higher duty cycle implies a relatively larger dc component in the Fourier series expansion and, as a consequence, a lower value of . Moreover, of in the case of current shapes – , the down slope of current shapes decreases with ; the lower slope implies that lower frequency components have increased amplitudes and this is a VITELLI: CALCULATION OF PER-UNIT-LENGTH RESISTANCE AND INTERNAL INDUCTANCE (a) 535 (b) (c) Fig. 9. f = 100 Hz. further cause of the decrease of versus . The values of obtained in the case of current shapes – are nearly coincident with , as it should be expected by considering the relatively large value of the dc component in the Fourier series expansion of such waveforms. The square markers shown in Fig. 2(a) and (b) indicates that obtained in the hypoththe results of the computation of esis that the current shapes are not replicated with period : and are evaluated just in the first period, during transient operation. That is, a given current waveform is turned on at and the time-domain solution is allowed ), to progress until the end of the first period is reached ( (inductance ) values whereupon the equivalent resistance are calculated. This kind of calculation can be greatly helpful in the analysis of ohmic losses in PFC converters, which gen– in erate current waveforms like those ones labeled by Table I. In such a case, not only is the frequency-domain approach particularly time expensive, but so is the time-domain to be adopted for the approach. In fact, the time-step size numerical integration of (11) must be chosen smaller than the (say ) and the time-domain switching period solution must be carried out until the steady-state is reached, that of mains periods. is, after an unpredictable number is 200 kHz; A typical value of switching frequency s, and s. This then, ms ( Hz), means that, since the main period is in order to get the steady-state solution in DOC, the time-domain approach would require the numerical solution of (11) in different instants. However, since current – can be seen as composed by sequences of waveforms different triangular waveforms that are characterized by unlike values of duty cycle and slope , an alternative way to follow is based on the preliminary numerical determination of the deon and for triangular current waveforms. pendence of Such a dependence, which can be obtained by carrying out an adequate number of numerical simulations adopting triangular with different values of and , can be current waveforms which inrepresented by means of the function terpolates the results of numerical simulations. Of course, the must be evaluated in correspondence of resistance value just the first period of each triangular current waveform (and is obtained, a not in steady-state DOC). Once function time-varying per-unit-length resistance can be associated to a given PFC current waveform where funcand represent the parameters characterizing the tions shape of the triangular waveform coincident with the PFC current at instant . Further work is in progress on this topic and will be the subject of a forthcoming paper. B. Sinusoidal Steady-State Analysis Such analysis has been carried out, as explained in Section II, by means of a FEM formulation of (7); the current injected into A , the rathe conductor is mil dius of the considered conductor (AWG 20) is m and its conductivity is S/m. In are reported (dashed Fig. 3, the results of computations of 536 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 44, NO. 4, NOVEMBER 2002 (a) (b) (c) (d) (e) Fig. 10. f = 1 MHz. line) and compared with theoretical predictions (solid line) as a function of frequency ; numerical findings and theoretical results are in very good agreement. First of all, it is useful to and the imfurther clarify the difference existing between ; they are nearly coincident at low frequenpressed current cies while they are completely different at higher frequencies as Hz, shown in Figs. 4–7 where the results obtained at kHz, MHz and MHz are, respectively, shown. In particular, the higher the frequency of the the lower, as expected, the forcing term due to the marked contribution, at higher freamplitude of quencies, of the integral term appearing in the right-hand side of (10). The role played by inductive phenomena, at higher frebut also on quencies, is evident not only on the amplitude of its shape. In fact, for example, at Hz (Fig. 4(b)) is practically coincident with its sinusoidal steady-state compowhich in turn is coincident with the impressed current nent (Fig. 4(a)); instead, at MHz, and at MHz, , a not negligible transient component is superimposed to as shown in Figs. 6(b) and 7(b), respectively. The current waveforms shown in Fig. 6(a) and (b), and in Fig. 7(a) and (b), clearly and that of reveal that the ratio between the amplitude of assumes large values at high frequencies. The most important motivation for the adoption of the time-domain approach to carry out sinusoidal steady-state analysis concerns the possibility to study the distribution of the current density during the whole period . This directly leads to the introduction of per-unit-length time-varying instantaneous pa- VITELLI: CALCULATION OF PER-UNIT-LENGTH RESISTANCE AND INTERNAL INDUCTANCE 537 rameters. In the following, we will focus on the instantaneous defined as per-unit-length resistance (16) is obtained as the ratio From (13.1) and (16), it is clear that between the average values, in a period , of the corresponding . quantities appearing in the numerator and denominator of The numerical results obtained clearly reveal a strong depenas a function of the frequency. As dence of the shape of examples, in Fig. 8(a) and (b), the computed waveforms of (thick line with circles at the computed points) for Hz MHz are respectively shown and compared with the and corresponding values of (thin line). At low frequencies is constant and equal to which is in turn equal to the dc value ; at high frequencies of resistance exhibits a nonmonotonic periodic behavior with a period equal to . In particular, in Fig. 8(a) and (b), the computed points have been marked with circles to stress the following fea: tures of , at high frequen• even relatively far from the zeros of assumes very high values; cies, , at low frequen• even relatively close to the zeros of assumes a constant value equal to . cies, Such features can be explained by analyzing the influence of frequency on the current density distribution in steady-state conditions. In Fig. 9(a)–(c), the spatial distributions of , taken at three equally-spaced time instants inside the period , for Hz, are reported; while in Fig. 10(a)–(e), the spatial distributions of , taken at five equally-spaced time inMHz, are shown. Two important differstants, for ences among the distributions of Fig. 9 and those of Fig. 10 are clearly evident. The first one is well known and is represented by the strong nonuniformity of the distributions of obtained for MHz as compared to the flat ones obtained for Hz. The second important difference is MHz, not only is far related to the fact that, for from being uniform on the conductor cross section, but the shape of its distribution strongly varies with time. In particular, for Hz, can be expressed with very good apwhile, for MHz, it proximation as , that is, it cannot be written cannot be expressed as ) times a funcas the product of a function of time only ( tion of spatial coordinates only ( ). This just means that , for MHz, changes within the pethe shape of riod while, at lower frequencies, it remains unchanged. In fact, for example, the transition from the distribution of Fig. 10(a) to that one of Fig. 10(c), both characterized by the concentration of current only in a narrow external ring of the conductor cross section, takes place through the intermediate distribution of Fig. 10(b). This last exhibits regions of the cross section with current density values of opposite sign and, moreover, internal rings characterized by higher absolute values of with respect to those ones flowing in more external rings. Such behavior can be qualitatively explained by modeling the whole conductor as (a) (b) Fig. 11. (a) i t(t) for p(t) = and r (thin line). 1(t). (b) r (t) (thick line) for p(t) = 1(t) a set of parallel wires, of very small cross section; on each elemental wire the current density can be considered almost uniformly distributed. The steady-state current carried by each one MHz. of such conductors is sinusoidal at a frequency However, due to inductive phenomena taking place, the currents in the internal wires are characterized by a phase lag with respect to the currents in the outer wires. That is why, as evident in Fig. 10(b), such currents do not change their sign synchronously; as a consequence, in correspondence of the current density distribution of Fig. 10(b), while the denominator of (16) is small, the numerator is relatively large thus leading to very high as shown in Fig. 8(b). Similar considerations also values of hold with reference to the distributions shown in Fig. 10(c)–(e). It is worth noting that the proposed formulation can be applied to isolated conductors with arbitrarily shaped cross sections. Since in modern electronic circuits conductors of rectangular cross section are widely adopted, the study of the in the case of isoinstantaneous patterns of current density lated rectangular conductors is of great interest but it is beyond the objectives of this paper. Contour plots of time-harmonic currents in rectangular conductors have been shown in [15]. As a further example of application of the proposed formu, where 1 is the Heaviside step lation the case function, has been also considered. In Fig. 11(a), the obtained is shown; while in Fig. 11(b) (thick line) is current 538 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 44, NO. 4, NOVEMBER 2002 reported and compared with the dc value (thin approaches line). As is evident, after an initial transient, . the dc value As a consequence of the above considerations it can be stated that the isolated conductor under study must be described, from a circuit point of view, by means of time-varying per-unit-length distributed parameters which are strongly dependent on the shape of the injected current. IV. CONCLUSION In this paper, a simple formulation based on the magnetic vector potential and on the scalar potential has been presented and applied to study skin effect current driven problems. The proposed method has been used to evaluate the equivalent per-unit-length resistance and internal inductance of conductors carrying typical SMPS distorted current waveforms. Timevarying per-unit-length parameters, which are strongly dependent on the shape of the injected current, can be introduced to describe the isolated conductor from a circuit point of view. [6] J. L. Maksiejewski, “Calculation of losses in conductors due to chopped impulse currents, taking the skin effect into account,” IEE Proc.-Sci. Meas. Technol., vol. 144, no. 3, pp. 111–116, 1997. [7] G. Antonini, A. Orlandi, and C. R. Paul, “Internal impedance of conductors of rectangular cross section,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 979–985, July 1999. [8] J. Guo, D. Kajfez, and A. W. Glisson, “Skin-effect resistance of rectangular strips,” Electron. Lett., vol. 33, no. 11, pp. 966–967, May 1997. [9] J. F. Imhoff, G. Meunier, X. Brunotte, and J. C. Sabonnadiere, “An original solution for unbounded electromagnetic 2-D and 3-D problems throughout the finite element method,” IEEE Trans. Magn., vol. 26, pp. 1659–1661, Sept. 1990. [10] P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineering. Cambridge, U.K.: Cambridge Univ. Press, 1996. [11] B. Carsten, “High frequency conductor losses in switchmode magnetics,” in Rec. HFPC ’86, Virginia Beach, VA, May 1986, pp. 155–176. [12] R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics. Norwell, MA: Kluwer, 2001. [13] D. C. Hamill, J. H. B. Deane, and J. P. Aston, “Some applications of chaos in power converters,” in Proc. IEE Colloq. Update on New Power Electronic Techniques (Digest No: 1997/091), 1997, pp. 5/1–5/5. [14] J. H. B. Deane and D. C. Hamill, “Improvement of power supply EMC by chaos,” Electron. Lett., vol. 32, no. 12, p. 1045, 1996. [15] D. Kajfez, J. Guo, and A. W. Glisson, “Magnetic field patterns inside strip conductors,” Microwave Opt. Technol. Lett., vol. 20, pp. 272–274, 1999. REFERENCES [1] C. R. Paul, Analysis of Multiconductor Transmission Lines. New York: Wiley, 1994. [2] O. Bíró, K. Preis, W. Renhart, G. Vrisk, and K. R. Richter, “Computation of 3-D current driven skin effect problems using a current vector potential,” IEEE Trans. Magn., vol. 29, pp. 1325–1328, Mar. 1993. [3] J. P. Webb, B. Forghani, and D. A. Lowther, “An approach to the solution of three-dimensional voltage driven and multiply connected Eddy current problems,” IEEE Trans. Magn., vol. 28, pp. 1193–1196, Mar. 1992. [4] O. Bíró, P. Böhm, K. Preis, and G. Wachutka, “Edge finite element analysis of transient skin effect problems,” IEEE Trans. Magn., vol. 36, pp. 835–839, July 2000. [5] S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics. New York: Wiley, 1984. Massimo Vitelli was born in Caserta, Italy, on July 20, 1967. He received the Laurea degree in electrical engineering from the University of Naples “Federico II”, Naples, Italy, in 1992. In 1994, he joined the Department of Information Engineering of the Second University of Naples, Naples, Italy, as a Researcher. In 2001, he was appointed Associate Professor in the Faculty of Engineering of the Second University of Naples where he teaches electrotechnics. His main research interests concern the electromagnetic characterization of new insulating and semiconducting materials for electrical applications, electromagnetic compatibility and the analysis and simulation of power electronic circuits.