Analyzing and Characterizing Common Mode Chokes for Three Phase Systems

4916 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 39, NO. 5, MAY 2024 Analyzing and Characterizing Common Mode Chokes for Three Phase Systems Álvaro Ojeda-Rodríguez , Carlos Domínguez-Palacios, Joaquín Bernal-Méndez , Senior Member, IEEE, and María A. Martín-Prats , Senior Member, IEEE Abstract—Common mode chokes (CMCs) are widely employed to reduce common mode emissions of three phase systems. This work presents an advanced technique for analyzing and characterizing CMCs with three or four windings. The central idea of the method is a modal analysis of the choke, where the modes are based on the modes of the noise signal transmitted by the three-phase line. These modes are defined following the criterion of its suitability from the perspective of electromagnetic interference control. This modal analysis is used to propose a circuit model of the choke that accounts for the impact of high-frequency parasitic effects and, at the same time, allows for a straightforward parameter identification. This parameter identification is achieved by using a characterization technique that makes use of measurements of the response of the CMC in simple connections, whose suitability is justified thanks to the ability of the modal analysis to provide analytical and easy-to-interpret expressions of these connections. This makes also possible to predict and quantify mode conversions caused by the device in the three-phases line. The accuracy of the obtained circuit model within a wide frequency range has been demonstrated by comparing measurements with calculated responses for several three- and four-wires CMCs. Index Terms—Common mode choke (CMC), conducted emissions attenuation, conducted emissions propagation modes, high frequency circuit model, modal analysis, parasitic effects. I. INTRODUCTION CURRENT trends in power electronics are toward the increase of switching frequencies and the power density of converters. An inevitable outcome of this tendency is the progressive increase of the electromagnetic emissions of power converters [1]. This poses a serious problem due to the necessity to comply with electromagnetic compatibility (EMC) regulations, which establish limits on the electromagnetic emissions of power converters in wide frequency ranges [2],[3],[4]. Manuscript received 1 August 2023; revised 10 October 2023 and 30 November 2023; accepted 21 January 2024. Date of publication 26 January 2024; date of current version 20 March 2024. This work was supported in part by the project TED2021-131954B-I00, funded by MCIN/AEI/10.13039/ 501100011033 and in part by the European Union “NextGenerationEU”/PRTR. Recommended for publication by Associate Editor Minjie Chen. (Corresponding author: Álvaro Ojeda-Rodríguez.) Álvaro Ojeda-Rodríguez, Carlos Domínguez-Palacios, and María A. Martín- Prats are with the Department of Ingeniería Electrónica, Universidad de Sevilla, 41092 Sevilla, Spain (e-mail: [email protected]; [email protected]; [email protected]). Joaquín Bernal-Méndez is with the Department of Física Aplicada III, Universidad de Sevilla, 41092 Sevilla, Spain (e-mail: [email protected]). This article has supplementary material provided by the authors and color versions of one or more figures available at https://doi.org/10.1109/TPEL.2024.3359092. Digital Object Identifier 10.1109/TPEL.2024.3359092 Fig. 1. Schematic representation of CMCs with different number of windings. (a) 2W-CMC. (b) 3W-CMC. (c) 4W-CMC. Electromagneticinterference(EMI)filtersareusedtomitigate electromagnetic emissions of electronic equipment [5]. Among the components of EMI filters, common mode chokes (CMC), which are made up of two or more coupled coils wounded on a magnetic core, are widely employed due to its ability to reduce common mode (CM) emissions [5],[6]. Fig. 1shows an sketch of a typical CMC used in one-phase systems. We will refer to this device as as two-wire CMC (2W- CMC). By contrast, three-phase systems can be implemented with either three-wire or four-wire conductors line [7],[8]. Consequently, CMCs used in three-phase systems are either three-wire CMCs (3W-CMCs) like that in Fig. 1(b), or four-wire CMCs (4W-CMCs) like that in Fig. 1(c) [8],[9]. In recent years, an important effort has been made to develop techniques for obtaining high-frequency circuit models of 2W- CMCs [10],[11],[12],[13],[14],[15],[16],[17],[18],[19]. However, much less works have been published dealing with the characterization of CMCs for three-phase applications or analyzing its effect on noise modes in the three-phase line [20].The current trend in modern power electronics toward the increase of switching frequencies has been boosted by recent advances in semiconductor technologies and by the upsurge of applications, suchasthoserelatedtotransportelectrification, where achieving high power density is a priority. This is expected to exacerbate EMC issues in modern three-phase systems [21],[22].Inthis context, a method capable of providing accurate broadband models of 3W-CMCs and 4W-CMCs can help to develop virtual prototypes of EMI filters and/or complete systems. This can reduce the time and cost associated with trial and error design processes, thus allowing the development of EMC-compliant three-phase systems with high power densities [12],[15],[23]. Design methods for 3W-CMCs have already been presented in [21] and [24]. However, those methods are more focused on © 2024 The Authors. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ OJEDA-RODRÍGUEZ et al.: ANALYZING AND CHARACTERIZING CMC FOR THREE PHASE SYSTEMS 4917 the design of 3W-CMCs to comply with certain attenuation requirements than on characterizing a given sample. As a consequence, they concentrate on modeling the CM impedance of the 3W-CMC,andadifferentialmode(DM)modelofthe3W-CMCs is not derived. Also, [12] reports a single example of application of a characterization method developed for 2W-CMCs to the case of a 3W-CMCs. However, the proposed circuit model is purely behavioral and the characterization technique is based on amore complexsetofmeasurements. Asimplercharacterization method for 3W-CMCs, requiring only a measurement with a quite simple setup, has been presented by Dominguez-Palacios et al. [25]. However, this method has a limited accuracy due to a nonprecise characterization of the effect of intrawinding coupling. Also, in that work, no complete modal theory is provided that can account for the response of the 3W-CMC in different connections and explain its effect on the modes of the line. In addition, the works cited above do not deal with 4W-CMCs. As we will see, the 4W-CMC case has particularities that do not allow us to treat it as a straightforward generalization of the 3W-CMC case. In this work, accurate wideband circuit models and a simple characterization technique are proposed for both 3W-CMCs and 4W-CMCs. The ideas, concepts, and experimental techniques developed in this work are supported by a modal analysis that allows a significant simplification of the analysis of the response of 3W- and 4W-CMCs, facilitating their characterization. To carry out this analysis, we start by defining a mode decomposition for signals in three-phase transmission lines (with three or four wires) that is convenient from an EMC perspective. We demonstrate that, given these modes, the condition that ensuresthata3W-CMCora4W-CMCdoesnotcauseconversion between modes is that the circuit model of the CMC has a set of natural modes that are derived from those defined in the line. We propose equivalent circuits for 3W-CMCs and 4W-CMCs that model them as six- or eight-port networks consisting of LCR resonant tanks with both magnetic and electrical coupling between them. It is shown that, under the reasonable assumption that the electrical couplings between coils are equal, the 3W-CMCs do not cause conversion between the modes defined in three-wire transmission lines. By contrast, the fact that the magnetic couplings between nearby and distant coils are different in 4W-CMCs causes conversion between the so-called differential and homopolar modes in four-wire lines. Moreover, we demonstrate that, for both 3W-CMCs and 4W-CMCs, it is possible to construct an alternative circuit model, derived from that originally proposed, that has the additional advantage that its parameters are grouped into blocks so that each block is associated with only one of the natural modes of the circuit. This allows for an easy assignment of values to the parameters of the model. To that end, we propose to use an advanced search algorithm that takes as input the measurements of a pair of transmission coefficients of the CMC in two simple setups, which reveal all the modal responses of the CMC. To demonstrate the suitability of the proposed circuit models and characterization method, we have characterized and verified a wide range of 3W- and 4W-CMCs. Mode conversion caused by TABLE I DEFINITION OF NOISE PROPAGATION MODES 4W-CMCs with symmetrically distributed windings around the core has also been experimentally verified. The rest of this article is organized as follows. Section II presents and justifies a convenient definition of modes for a noise signal propagating in a 3-wire or 4-wire three-phase line. Section III proposes high-frequency circuit models for 3W- and 4W-CMCs and presents the key ideas of a generalized modal analysisthatfacilitatesanalysisof the problemandidentification of the elements of the circuit models. Section IV outlines a particularly simple technique to find the value of the parameters of the circuit models of the CMCs. A method to validate the equivalent circuit models of 3W- and 4W-CMCs is presented in Section V. Section VI is devoted to presenting several examples to demonstrate the precision and evaluate the scope of the characterization technique for 3W- and 4W-CMCs proposed in this work. Finally, Section VII concludes this article. In addition, in Appendix A, a mathematical demonstration is provided for a result with high relevance in the context of this work. II. NOISE MODES DEFINITION Three-phase power lines can be regarded as (N+1)- conductors transmission lines with N=3,orN=4if neutral conductor is present [26]. By comparison, single-phase transmission lines (with N=2) support in general two propagation modes. In the analysis of conducted emissions in single-phase systems, they are typically decomposed into CM and DM. This is done for convenience, as CM and DM typically have different generation mechanisms, give rise to different electromagnetic compatibility issues (e.g., CM emissions tend to more easily create coupling problems and radiated emissions), and require specific attenuation techniques [5]. Therefore, it is reasonable to apply the same criteria in determining which propagation modes should be defined in three-phase systems. Distinction between CM and DM in three-phase systems has been carried out in several previous works, mostly in three-wire systems [20],[27],[28]. In a few studies, this decomposition is performed in four-wire lines, but exclusively to study the behavior of CM, whose definition may vary [29],[30].Insummary, an agreement cannot be found in the literature on the more convenient definition, from an EMC perspective, of propagation modes for three-phase systems. To address this problem, in this work, we propose the decomposition of the signal in a four-wire line according to the modes shown in Table I. The currents associated with each mode are shown in Fig. 2. The idea behind this decomposition is to obtain a decomposition of the signal on 4918 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 39, NO. 5, MAY 2024 Fig. 2. Representation of noise propagation modes defined in this work for four-wire three phase systems. thelineintotwomodes unrelated to the functionalcurrentsofthe system, labeled as CMs (CM1 and CM2) in Table I, and another two modes, identified as DMs (DM1 and DM2), which account for the (low-frequency) power currents that must flow through the line. The CM1 and CM2 modes in Table Iare identified as CMbecause theyimplya nonzeronet currentflowinginthethree power lines (lines a, b, and c in Fig. 2). In fact, CM1 is a true CMforthe four-wiresline, whereasCM2standsfor a homopolar or zero-sequence component of the three-phase current. These modes CM1 and CM2 are similar to CM defined in singlephase lines in that they have specific generation mechanisms in power electronics devices attached to the power line, which are typically related to the presence of parasitic capacitances in nodes with rapid voltage variations [5]. In addition, these CM modes can easily give rise to coupling and radiation problems and must be attenuated using specific techniques, including the use of CMCs. The modal decomposition shown in Table Iis an extension to four wires of the proposal reported in [20],[28] for three-wire systems. From a mathematical point of view, it is more convenient to normalize the orthogonal voltage vectors in Table Iso that they all have a unit magnitude. This normalization leads to the following transformation matrix: T4W = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 1 2√3 1 √6 1 √2 1 2 1 2√3 1 √6−1 √2 1 2 1 2√3−2 √60 1 2−3 2√300 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (1) This modal decomposition coincides with the natural modes of a sufficiently symmetric four-wires line [26]. Note that the columns in T4W are the normalized CM1, CM2, DM1, and DM2modesdefinedinTableI.Thus,T4W representsthematrix of change of basis between the (normalized) modes defined in Table Iand the ordinary basis of the line. The modal decomposition given in (1) can be readily particularized for a three-wire transmission line by using the following transformation matrix: T3W= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 √3 1 √6 1 √2 1 √3 1 √6−1 √2 1 √3−2 √60 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2) This 3×3transformation matrix can be obtained simply by eliminating the first column and the last row of T4W in (1) and performing a convenient renormalization of the column vectors. Note that the modal decomposition given by T3W in (2) corresponds of the Clarke or alpha-beta transformation commonlyusedin three-phasesystems[31],[32].FromanEMC perspective, the first column in T3Wcorresponds to a CM, and theothertwocolumnsaretheDMsthataccountforthefunctional currents that are expected to flow in the line. III. MODAL ANALYSIS AND HF CIRCUIT MODEL In general, a CMC or any other filtering device inserted in aN+1-conductors transmission line should be regarded as a 2N-ports passive network. In single-phase systems, the effect of the CMC is analyzed by studying the insertion loss for each mode (CM and DM) of the line [5]. However, asymmetries in the filteringdevicecan result in energytransfer betweenmodes[33]. Regarding this, Ojeda-Rodríguez et al. [19] demonstrated that the condition for a 2W-CMC not to cause undesired mode conversion between CM and DM in a single-phase transmission line translates into a necessary and sufficient condition for the four natural modes (eigenvectors of the admittance matrix) of the 2W-CMC. Therefore, a pertinent question is whether, for a set of Npropagation modes, such as that defined in the previous section for a three-phase transmission line, there also exists a set of2Nnaturalmodesof the CMC that ensuresthatthedevicewill causenomodeconversionintheline.Theanswertothis question is provided in Appendix A, where it is demonstrated that for a set of Ntransmission modes selected in an N+1-conductors line and organized as columns vectors of an N×Nmatrix (T), the condition that a passive device with input–output symmetry (such as a CMC) inserted into that line must satisfy in order to avoid mode conversion between those Ntransmission modes is that the 2Nnatural modes of its 2N-ports circuit model can be arranged as column vectors of a 2N×2Nmatrix (E), which can be expressed in terms of Tsubmatrices as follows: E=TT T−T.(3) Therefore, Ein (3) establishes the necessary relationship that must exist between the Npropagation modes of the line and the 2Nnatural modes of the CMC (or any other 2N-ports device inserted in the line) that ensures no mode conversion among these Npropagation modes on the line. A. Circuit Model for a 3W-CMC Essentially, a 3W-CMC consists of three tightly coupled coils, and its objective is to attenuate the propagation of the CM on the three-wire line without causing conversion between this mode and the DMs that can propagate in the line. This is achieved in principle by a simple arrangement consisting of three equal coils with a strong mutual coupling between them. However, parasitic capacitances between turns cause each of these coils to behave actually as parallel LCR resonators [11],[12],[17]. In addition, electrical coupling between different coils must be OJEDA-RODRÍGUEZ et al.: ANALYZING AND CHARACTERIZING CMC FOR THREE PHASE SYSTEMS 4919 Fig. 3. Circuit model for the 3W-CMC with three evenly distributed windings. Each winding is modeled by a LCR resonant circuit. Mand YWaccount for the magnetic and electric couplings between windings, respectively. TABLE II MODES OF A SIX-PORTS NETWORK (3W-CMC)ENSURING NOMODE CONVERSION BETWEEN MODES DEFINED IN A THREE-WIRES LINE accounted for in order to achieve a wideband circuit model [11]. Those effects are schematically represented in Fig. 3. This figure also includes a proposed circuit model for the 3W-CMC, which is a six-ports network thoroughly devised to account for these high frequency effects and, at the same time, to ensure that the 3W-CMC does not cause mode conversion of the modes defined in (2) for three-wire transmission lines, which is a desirable characteristic of practical 3W-CMCs. To demonstrate that the circuit model of the 3W-CMC in Fig. 3will not give rise to mode conversion between the propagation modes on the three-wire lines defined by T3W in (2) it is enough to verify that the eigenvectors of the admittance matrix of that six-ports network coincides with those listed in Table II, which are obtained by applying the splitting shown in (3) to the set of modes defined by T3W. Note that the voltage modes in Table II are not normalized to make them easier to read and interpret.ItisinterestingtonotethatthefirstthreemodesinTable II (i.e. modes G, W1, and W2) produce equal excitations at both terminals of each coil. Therefore, these modes account for the capacitive responses of the 3W-CMC. On the other hand, modes C, D1, and D2 in Table II involve a voltage drop between the terminals of the inductances of the 3W-CMC. However, while the C mode will generate CM currents that sum their magnetic Fig. 4. Modal-parameters circuit (MPC) model for the 3W-CMC. fluxes in the core of the 3W-CMC, modes D1 and D2 will result in differential currents in the 3W-CMC, i.e., currents whose net magnetic flux in the core of the 3W-CMC would be nearly zero. Note that all the elements of the circuit model in Fig. 3 have a clear physical meaning. However, this circuit model does not comply with additional requirements, which would also be highly desirable. That is, that the parameters of the circuit model can be independently associated with the modal admittances (eigenvalues of the admittance matrix) of the 3W- CMC. This feature is essential because it greatly simplifies the extraction of parameters for the circuit model of the 3W-CMC, as demonstrated in [19] for the case of 2W-CMCs. In that work, the proposed circuit model for 2W-CMCs is referred to as the modal parameters circuit (MPC) of the 2W-CMC. Therefore, the question immediately arises whether it is possible to modify the circuit in Fig. 3to construct an MPC model for 3W-CMCs. Fig. 4shows a MPC model of a 3W-CMC that is completely equivalenttothatinFig.3.Thecomponents oftheMPCmodelof the 3W-CMC in Fig. 4are arranged in two blocks: asymmetrical mode test block (left) and symmetrical mode test block (right). Asymmetrical mode test block is made up of three perfectlycoupled windings (kc=1), each one modeled by the inductance LC/3, where LC=L+2Mis the inductance seen by the CM current.Thetypicalresonantbehavior of thesecoilsisaccounted for by the parallel CCand RCelements. Also, the YRLC admittance in the asymmetrical mode test block, consisting of one or more parallel RC and/or RL branches, is intended to take into account the effects of either magnetic losses or dielectric losses,asexplainedin [19]. Notethat,dueto the perfectcoupling between coils in the asymmetrical mode test block, this block will be short-circuited by current modes other than C. On the other hand, symmetrical mode test block is made up of three pairs of inductances LD/6coupled as indicated in Fig. 4, being kd=1and LD=L−M, where LDis the inductance seen by 4920 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 39, NO. 5, MAY 2024 TABLE III MODAL ADMITTANCES OF THE MPC MODELOFA3W-CMC IN FIG.4 differential currents. It can be checked that this coupling forces inductances of the symmetrical mode block to be short-circuited by both common (C) and intra-windings (W1, W2) modes. In this block, CDmodels the capacitive behavior of the windings at high frequencies when excited with DMs D1 and D2, whereas YWdetermines the response of the 3W-CMC when excited with intra-windings modes W1 and W2. This YWadmittance will be, in principle, that of a simple CWinter-winding capacitance. However, if necessary, the YWadmittance can also integrate a LCRWnetwork (a parallel LCR tank) in series with the CW inter-winding capacitance. As explained in [19], this resonant tank accounts for inter-turns resonances, which arise in some cases at sufficiently high frequencies due to the fact that the wavelength inside a core with high permeability and high permittivity may become quite small [34]. Finally, it should be noted that although a parasitic capacitance between each single terminal and ground Cgis included in Fig. 4, its effect can be neglected in this analysis for the same reasons as explained in [19]. Itcanbeeasilycheckedthattheeigenvectorsoftheadmittance matrix of the 3W-CMC MPC circuit model in Fig. 4are those listed in Table II. A first hint of the suitability of the proposed circuit model comes from the simplicity of the corresponding eigenvalues or modal admittances, which are given in Table III. Moreover, note that none of the parameters of the model is involvedinmorethanonemodaladmittance.Thisisakeyfeature of the proposed MPC model that, along with the simple measurement setups that will be presented in Section IV, facilitates the efficient determination of its parameters. B. Circuit Model for a 4W-CMC Fig. 5shows the main physical effects that should be modeled for a 4W-CMC manufactured with a toroidal core and four evenly distributed windings on it. In principle, the circuit model in Fig. 5is a direct extension of the 3W-CMC case to include one more LCR resonant circuit. However, a key difference between 4W-CMCs and 3W-CMCs is that in the latter case the relative position of each two windings in the core is the same, which allows us to assume that both magnetic and electric couplings between windings are the same for every pair of windings. However, a 4W-CMC has two different relative positions between Fig. 5. Circuit model for the 4W-CMC with four windings evenly distributed. Each winding is modeled by an LCR resonant circuit. M1and M2account for the magnetic coupling between nearby and distant windings respectively. The admittance YWaccounts for the electric coupling between windings. each pair of windings: either they are nearby windings or they are distant windings, i.e., placed at opposite sides of the ring core. For this reason, in the 4W-CMC model we will assume that the magnetic couplings between nearby coils are greater than between distant coils, i.e. M1>M 2. This difference in the magnetic couplings of different pairs of coils wound in the same core has been previously pointed out in [35], and it should be caused by the fact that the number of magnetic field lines linking both windings depends on the relative position of both windings in the core [6]. This effect will be experimentally verified for 4W-CMCs with symmetrically arranged windings in Section VI. In this section, its relevant practical consequences will be analyzed. Unlike magnetic couplings, the electric couplings between each pair of coils are assumed to be the same in the circuit model proposed in Fig. 5. This is justified by the fact that most core materials, for example MnZn or nanocrystalline cores, have extremely large electric permittivities [22],[34]. This results in strong electric couplings between the windings which is associated with the electric field that connects both windings throughthecore[18],[22].Sinceachangeintherelativeposition betweentwowindingsonatoroidalcoreimpliesthatthedistance betweenthemincreasesononesidebutdecreasesontheopposite side, the relative position of the two windings has little effect on the parasitic capacitance between them. This has been experimentally tested in [18], and we double checked it by measuring parasitic capacitances between different windings in 4W-CMCs with different core materials. Additional verification is provided by the good results provided systematically by the proposed circuit model to characterize a great number of different CMCs, as we will see in Section VI. 1) Mode Conversion in 4W-CMCs: In this section we will demonstrate that the difference between the mutual inductances of nearby and distant coils in the circuit model in Fig. 5causes mode conversion between some of the modes defined for a fourwireslineinTableI. OJEDA-RODRÍGUEZ et al.: ANALYZING AND CHARACTERIZING CMC FOR THREE PHASE SYSTEMS 4921 TABLE IV MODES OF A EIGHT-PORTS NETWORK (4W-CMC)ENSURING NOMODE CONVERSION BETWEEN MODES DEFINED IN A FOUR-WIRES LINE According to the demonstration presented in appendix A,the condition to ensure that the 4W-CMC does not cause mode conversionbetweenthelinemodesinTableI,isthattheeigenvectors of the 8×8admittance matrix of its circuit model, Y4W,are the columns of the following matrix: E4W =T4W T4W T4W −T4W(4) where T4W is defined in (1). The non-normalized voltage excitations defined by the columns of E4W are represented in Table IV. The first four rows in Table IV define excitations with the same voltage imposed at the two terminals of each winding and, therefore, are expected to give rise to a capacitive response of the device mainly associated with electrical coupling between windings. By contrast, the last four rows of Table IV correspond to voltage excitation where the windings are excited either in CM, in homopolar mode or in two additional DMs referred to here as D1 and D2. Note that if these modes happen to be eigenvectors of the admittance matrix of the 4W-CMC, each of themwillgiverise to currents directlyproportionaltothe voltage excitation represented by its eigenvector. Therefore, referring to the line modes in Table I, excitation of the 4W-CMC in mode C willresultinaCM1modeintheline,whereasexcitationinmode H will result in a CM2 mode propagating in the four-wires line, which is regarded as a zero-sequence or homopolar mode. This homopolar mode is a CM with respect to the three power lines (i.e., leaving the neutral cable out) but it is a DM with respect to the four wires because the sum of the excitations on these wires is zero. On the other hand, note that D1 and D2 are DMs with respect to either the three power lines or the four wires. As explained previously, D1 and D2 should be components of the functional currents of a three-phase device, which makes them equivalent to the DM defined in single-phase systems. A convenient way to analyze mode conversion in the 4W- CMC is to represent its admittance matrix Y4W on the basis given by E4W. In the case of the circuit model proposed in Fig. 5, this change of basis leads to the following result: Y4WE =E−1 4WY4WE4W =YUP 0 0YDW (5) Where 0stand for a 4×4zero matrix and YUP is a diagonal 4×4matrix that can be written as follows: YUP =⎡ ⎢ ⎢ ⎢ ⎣ YG000 0YW00 00YW0 000YW ⎤ ⎥ ⎥ ⎥ ⎦ (6) where YGand YWare capacitive admittances that account for electric couplings to the ground and between coils. On the other hand, YDW is another 4×4matrix that has the following form: YDW = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ YC000 0YDN1+2YDN2 3 YDN2−YDN1 3√2 YDN1−YDN2 √6 0YDN2−YDN1 3√2 YDN1+5YDN2 6 YDN2−YDN1 3√2 0YDN1−YDN2 √6 YDN2−YDN1 3√2 YDN1+YDN2 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (7) where YC,YDN1, and YDN2 are the eigenvalues associated with the four resonant modes (or eigenvectors) of the system of four magnetically coupled resonators that made up the 4W-CMC. Those resonant modes are illustrated in Fig. 6. It can be easily demonstrated that for four equal LCR resonators, those eigenvalues can be expressed as: YC=2jωCC+1 jωLC +1 RC(8a) YDN1 =2jωCD+1 jωLD1 +1 RD1 (8b) YDN3 =YDN2 =2jωCD+1 jωLD2 +1 RD2 (8c) where in principle, RC=RD1 =RD2 and CC=CDare the resistance and capacitance of the four equal resonators. We use here different subscripts for convenience, since the circuit model that will be presented in the next section will allow us to independently fix these parameters. Note that there is no difference between DN2 and DN3 modes from a physical point of view, since they provoke the same response of the 4W-CMC. Fromamathematicalpointofview,theyarelinearlyindependent eigenvectors of the admittance matrix with the same eigenvalue (degenerate eigenvectors). Note that the matrix YDW in (7) is not in general a diagonal matrix unless YDN1 =YDN2 (i.e., M1=M2). Therefore, Y4WE in (5) is not in general diagonal. In fact, the distribution of the elements of Y4WE reveals that the main effect of the inequality of mutual couplings between nearby and distant windings in a 4W-CMC is that, when inserted in a four-wires line, the 4W- CMC will cause mode conversion between the homopolar mode (CM2 in Table I) and the two DMs (DM1 and DM2) of the line. On the other hand, the fact that YCis the only nonzero element in its row and column in Y4WE indicates that the 4W-CMC can provide attenuation to the CM (CM1 mode in Table I) without causing mode conversion with the rest of the modes of the line. In summary, the problem of mode conversion in 4W-CMC can be understood by considering that the C, DN1, DN2 and 4922 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 39, NO. 5, MAY 2024 Fig. 6. Representation of the four natural modes of four magnetically coupled LCR resonators with different mutual inductances between nearby windings and distant windings (M1and M2). The effective inductances seen for each mode at the terminals of the windings of the 4W-CMC are indicated below. DN3 modes represented in Fig. 6are the four noncapacitive natural modes for a 4W-CMC with M1=M2. However, this set of modes is different from the noncapacitive modes (C, H, D1, and D2) listed in Table IV, which have been defined in (4) based on the criterion of their suitability from the EMC perspective. In particular, note that whereas C mode is present in both mode sets and, therefore, it is a natural mode of the 4W-CMC, the H, D1, and D2 modes in Table IV are not natural modes of the 4W-CMC unless M1=M2. From a practical point of view, this implies that the presence of the 4W-CMC in the three-phases line can give rise to a homopolar (H) noise component due to mode conversion from DMs noise and vice versa. 2) MPC Circuit Model for a 4W-CMC: In this section, we demonstrate that it is possible to find a MPC model for a 4W- CMC, in a similar fashion as those found for 2W-CMCs in [19] and for 3W-CMCs in Section III-A. This MPC model should be equivalent to that in Fig. 5, but must be made up of independent circuit blocks, each one accounting for the response of the CMC to only one of the distinct modes listed in Table IV. A careful analysis and decomposition of the circuit model in Fig. 5leads to the MPC circuit model of the 4W-CMC shown in Fig. 7. Once again, this MPC circuit model features two main blocks. The asymmetrical mode test block is made up of four magnetically and perfectly coupled (kc=1)LCR resonant circuits, so that these components are responsible for the response of the 4W-CMC to asymmetrical mode test excitation exclusively. Therefore, this block causes no effect in differential currents and currents from one winding to another through interwinding capacitances. The inductance of each winding is the inductance seen by a CM current, LC=L+2M1+M2. The role of admittance YRLC in this block is explained in Section III-A. As for the symmetrical mode test block, it is made up of two groups of four resonant circuits that are short-circuited by currents in C mode, and therefore account for the response of the 4W-CMC to differential excitations, and it also includes YWthat accounts for the electric coupling between windings. Regarding the resonators, the first group of them features four perfectly coupled inductances (kd1 =1), and it accounts for the resonance associated with the component of the DM current in theDN1modeillustrated in Fig. 6(b). Therefore, this component of the DM current sees an inductance LD1 =L−2M1+M2 Fig. 7. Modal-parameters circuit (MPC) model for a 4W-CMC. at low frequencies and a capacitance CDafter resonance. In addition,this groupofresonators isshort-circuitedby thecurrent modes DN2 and DN3 illustrated in Fig. 6(c) and (d).The second group of resonators within this symmetrical test block, characterized by LD2 =L−M2,CDand RD2 is designed to be short-circuited by both C mode currents and the DN1 mode currentsrepresentedinFig.6(b).Forthisreason,distantinductances are perfectly coupled (kd2a =kd2b =1). All the resonators that belong to the symmetrical mode block are short-circuited by any of the intra-winding modes (W1, W2 and W3) too. Moreover, the inductances of this second group of resonators are split to provide a connection point for the intrawinding admittance YW. This admittance is excited exclusively by the intrawinding modes W1, W2, and W3, which short-circuit all the resonant tanks in the circuit. Table Vlists the elements of the main OJEDA-RODRÍGUEZ et al.: ANALYZING AND CHARACTERIZING CMC FOR THREE PHASE SYSTEMS 4923 TABLE V DIAGONAL ELEMENTS OF MATRIX Y4WE IN (5) FOR THE MPC MODEL OF A 4W-CMC IN FIG.7 diagonalofY4WE intermsofthecomponentsoftheMPCmodel in Fig. 7. In summary, the MPC circuit model of the 4W-CMC in Fig. 7offers greater flexibility and independence of its parameters at the cost of slightly increased complexity compared to its equivalent circuit in Fig. 5. The main purpose of this trade-off is to significantly facilitate the process of determining thecircuit’s parameters and, consequently,improvethe accuracy of the characterization of the 4W-CMC. IV. CHARACTERIZATION TECHNIQUE The procedure proposed in this work to obtain a numerical estimate of the parameters of the MPC model for the 3W- and 4W-CMC is based on simple measurements along with the use of an advanced search algorithm. Since this approach is very similar to that already proposed in [19], we will only outline it here for the sake of brevity. The proposed characterization technique has followingtwo main steps: A. Step 1 First, the response of the CMC is measured in two quite simple connections which are referred to as open circuit (OC) connection, shown in Fig. 8, and the intrawinding (WD) connection, shown in Fig. 9. Apart from their simplicity, the key feature that makes these two connections specially suitable for characterizing the CMC is the fact that they explicitly reveal all the modal responses of the CMC. To demonstrate this, (9) and (10) show the transmission coefficients of the 3W-CMC and 4W-CMC in those two setups calculated in terms of modal admittances SOC 21 =3RYCYD YD+YC(2 + 3RYD)(9a) Fig. 8. Open circuit (OC) connection. (a) 3W-CMC. (b) 4W-CMC. In these schematics TG stands for tracking generator, and SA stands for spectrum analyzer. Fig. 9. Capacitive intra-winding (WD) connection. (a) 3W-CMC. (b) 4W- CMC. In these schematics TG stands for tracking generator, and SA stands for spectrum analyzer. SWD 21 =2RYDYW YD+YW+2RYDYW (9b) SOC 21 = 4RYCYDN1YDN2 2YCYDN1 +YCYDN2 +YDN1YDN2 4RYCYDN1YDN2 2YCYDN1 +YCYDN2 +YDN1YDN2 +1 (10a) SWD−near 21 =2RYDN1YW YDN1 +YW+2RYDN1YW (10b) SWD−far 21 =2RYDN2YW YDN2 +YW+2RYDN2YW .(10c) Note that SOC 21 in (9a) and (10a) are proportional to all the resonant modal impedances of each CMC. Therefore, we expect to measure two resonances (related to YCand YDin Table III) in the insertion loss of the OC curve of a 3W-CMC, while a 4W-CMCshould presentthreeresonancesrelatedtotheresonant admittances YC,YDN1 and YDN2 defined in (8). In the case of the WD connections, the response of the CMC should be proportional to the resonant differential admittances and to the admittance YW, which must be mainly capacitive. Considering that the main goal of WD measurement is to extract accurate information about YW, either SWD−near 21 or SWD−far 21 could be equally used to this end. In this work, we have used the measurement of SWD−far 21 for parameter extraction. B. Step 2 Since we have assigned a physically meaningful basic circuit structure to the modal admittances of the CMC that appear in (9) and (10), the task of assigning values to these parameters for a particular CMC is therefore reduced to find the set of parameters that provides a better agreement between the measured curves of 4924 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 39, NO. 5, MAY 2024 TABLE VI TYPICAL VALUES OF THE PSO ALGORITHM SETTING PARAMETERS the transmission coefficients in the OC and WD connections and the analytical expressions of these two transmission coefficients in (9) (for 3W-CMCs) or (10) (for 4W-CMCs). In this work, a particle swarm optimization (PSO) search algorithm has been used to this end [36]. This algorithm greatly benefits from the use of analytical expressions for the transmission coefficients and from the fact that the parameters of the MPC circuit model are independently associated with the different natural modes of the CMC. As a result, the typical computational time required to achieve a goodfitrangesfromafewsecondstoafewminutes. To expedite the replicability of the results obtained with the method proposed in this work, we provide typical values for the setting parameters utilized in the PSO algorithm in Table VI. It is worth mentioning that, to ensure a fast convergence of the search algorithm, it is convenient to provide suitable initial values for the parameters of the circuit model of the CMC. To this end, the self-inductance of a winding of the CMC can be measured with an LCR-meter, both in OC and in short-circuit configurations. This can be used along with the resonance frequencies, identified as sharp dips in the measured |SOC 21 |curves, to estimate the parasitic capacitances of the circuit model. Also, the magnitude of the transmission coefficient SOC 21 at the resonance frequencies can be used to estimate resistances in the LCR resonators. A detailed flowchart of the characterization technique outlined in this section can be seen in Fig. 6 within [19]. V. MODEL VERIFICATION In general, assessing the feasibility and accuracy of a circuit model requires performing measurements to compare the actual response of the device with the prediction provided by the circuit model. In this work, we will check the accuracy of the models of 3W- and 4W-CMCs by using the standard tests defined in CISPR-17 to measure the interference suppression characteristics of passive filtering devices [37]. An schematic of these tests for characterizing 3W-CMCs is shown in Fig. 10. The test in Fig. 10(a) is referred to as asymmetrical mode test and it is intended to determine suppression of CM. The test in Fig. 10(b) is referred to as symmetrical mode test and allows measuring the response of the device to DMs. Fig. 10. CISPR-17 standard connections to characterize the insertion loss of a symmetric filtering device with six ports. (a) Asymmetrical mode or CM measurement setup. (b) Symmetrical mode or DM measurement setup. By using the modal analysis presented in Section III,itis possible to obtain expressions for the transmission coefficients for asymmetrical and symmetrical mode tests in Fig. 10 (Sasym 21 and Ssym 21 ) in terms of the modal admittances of the CMC. This provides a quick understanding about what these setups actually measure.Forthecaseofthe3W-CMC,thefollowingexpressions are obtained: Sasym 21 =2R(YC−YG) (2RYC+ 1)(2RYG+1) (11) Ssym 21 =2R(YD−YW) (RYD+2)(RYW+2).(12) Intheseequations,itcanbeobservedthattheresponseofthe3W- CMCtotheasymmetricalmodetestisdeterminedbyYCandYG, whereas the response to symmetrical mode test is determined by YDand YW. These expressions are identical to those obtained for the 2W-CMC, [19]. Regarding the 4W-CMC, note that the transmission coefficient measured in the asymmetrical mode test setup is the same as that already provided for 3W-CMCs in (11). Thus, it can be deduced that, provided that proper care is taken in the measurements by putting the CMC sufficiently away from grounded surfaces, the asymmetrical mode test measures the response of the C mode of the 3W- or 4W-CMC, i.e., its attenuation of the CM in the three-wires or four-wires line. However, for a 4W-CMC with different mutual inductance coefficients for nearby and distant windings, two different expressions of the transmission coefficients for the symmetrical mode test can be obtained depending on whether the measurement involves two nearby coils or two distant coils Ssym−far 21 =2R(YDN2 −YW) (RYDN2 +2)(RYW+2) (13) Ssym−near 21 = 2RYDN1YDN2 YDN1 +YDN2 −RYW RYDN1YDN2 YDN1 +YDN2 +1 (RYW+2) .(14) OJEDA-RODRÍGUEZ et al.: ANALYZING AND CHARACTERIZING CMC FOR THREE PHASE SYSTEMS 4931 The mixed-mode S-parameter matrix of a reciprocal device with input/output symmetry can in general be expressed as follows: Smm =10 01 ⊗Si+01 10 ⊗Sj(26) where the terms in Siare the reflection coefficients of the N modes of the line and the terms in Sjare the transmission coefficients.Thus,the keycondition that ensuresno mode conversion in the line is that both Siand Sjare purely diagonal matrices. Let therefore first assume that the columns of Ein (20) are eigenvalues of the admittance matrix of the 2N-network and let demonstrate that this is a sufficient condition to ensure no mode conversion between the modes defined by Tin the transmission line. Since Eis actually a change of basis matrix from the basis of the eigenvalues to the terminals basis, this can be used to calculate the admittance matrix in the basis of its eigenvectors as: Ye=E−1·Y·E. Since the vectors in Eare eigenvectors ofY, this meansthatthe admittancematrixYemust bediagonal, being the terms of the diagonal the modal admittances (or eigenvalues). The mixed-mode S-parameter matrix of the device can then be calculated by firstly calculating the S-parameter matrix in the modal basis, Se,fromYeusing (23) and then carrying out a change of basis form the basis of the eigenvectors to the mixed-mode basis Smm =M−1·E·Se·E−1·M(27) where M−1·E=11 1−1⊗I(28) E−1·M=(M−1·E)−1=1 211 1−1⊗I.(29) By assuming that Seis diagonal, and making use of the basic rules of the algebra of the Kronecker product [44], it can be demonstrated that Smm has the form described in (26), where Siand Sjare diagonal matrices. This allows us to conclude that this 2N-ports network does not cause conversion between the modes defined in the line. It is also possible to demonstrate that the fact that the columns of Ein (20) are eigenvalues of the admittance matrix of the 2N-network is a necessary condition to ensure that no mode conversion exists in the transmission line. 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