Modal Theory and Approach for Accurate Characterization of Common-Mode Chokes

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. -, NO. -, MONTH 2023 1 Modal Theory and Approach for Accurate Characterization of Common Mode Chokes A. Ojeda-Rodr´ ıguez, J. Bernal-M´ endez, Senior Member, IEEE, and M. A. Mart´ ın-Prats, Senior Member, IEEE Abstract—This work presents a method for characterization of common mode chokes, aimed at constructing accurate virtual prototypes of electromagnetic interference filters. This method is based on a general modal analysis that identifies the natural modes of a symmetric four-ports network in a power line. Natural modes excited in the setups defined in the CISPR-17 norm to characterize filtering devices are determined. From this analysis, two simple measurement setups particularly suitable for characterizing common mode chokes are identified, and a decisionmaking algorithm is presented to determine the parameters of a configurable circuit model of the common mode choke. This circuit model is especially devised so that each one of its different circuit blocks can be associated with a single modal response of the common mode choke. This, along with the use of physical criteria to define the basic schematic of each circuit block, ensures the simplicity, accuracy, and generality of the final circuit model. Characterization of many commercial common mode chokes with different core materials and winding configurations has been carried out to demonstrate that this circuit model is able to accurately account for the actual response of different types of common mode chokes in a wide frequency range. I. INTRODUCTION CURRENT trends in power converters are towards high power densities, especially in certain applications such as aeronautical, aerospace and automotive industries. For this reason, wide-bandgap (WBG) semiconductor devices, that allow increasing switching frequency and reducing size, are being widely adopted [1]. Due to this, electromagnetic compatibility (EMC) issues are becoming more important [2], [3]. To ensure proper operation of the systems, EMC standards typically establish limits to electromagnetic emissions, both radiated and conducted [4]–[8]. Electromagnetic interference (EMI) filters are widely employed to mitigate electromagnetic emissions of electronic equipment [9]. Most EMI filters incorporate one or more common mode chokes (CMC), which, for single-phase power lines, are made up of two highly coupled windings on a magnetic core. Although CMCs are mainly intended to attenuate common mode (CM) noise, they also provide attenuation for differential mode (DM) noise [2], [9], [10]. A. Ojeda-Rodr´ ıguez and M. A. Mart´ ın-Prats are with the Department of Electronic Engineering, University of Seville. J. Bernal-M´ endez is with the Department of Applied Physics III, University of Seville. This work has been partially supported by the ”Fondo Europeo de Desarrollo Regional” (FEDER) and the ”Consejer´ ıa de Transformaci´ on Econ´ omica, Industria, Conocimiento y Universidades de la Junta de Andaluc´ ıa”, Programa Operativo FEDER 2014-2020. project: US-1381111. This work has been partially supported by TED2021-131954B-I00 project, funded by MCIN/AEI/10.13039/501100011033 and “NextGenerationEU”/PRTR. There are multiple reasons driving a growing interest in the development of accurate wideband models for EMI filters and their components. One primary reason is the already commented trend towards the use of higher switching frequencies in power converters, motivated by the need to achieve higher power densities. This trend is particularly relevant in fields such as the electrification of means of transport, where there is a pressing need to meet stringent EMC requirements that cover a broad ranges of frequencies (e.g., 150 kHz to 152 MHz in section 21 of DO-160G for conducted emissions [7]), while simultaneously dealing with strict weight and volume restrictions [11]. In this context, increasing the accuracy of the model of the filter’s components is crucial since it allows to optimize the filter by avoiding the increase of weight and volume associated with over-design. In general, accurate circuit models of EMI filters are difficult to obtain because the high-frequency response of the filter is highly affected by parasitic effects [12]. The task of obtaining an accurate circuit model of a CMC is further complicated due to increasing trends toward the use of high-permeability magnetic cores, such as MnZn ferrite and nanocrystalline alloys [13]. These materials typically have very high permittivity values, which could make dominant the electric field inside the core in the overall electric coupling between turns [14], [15]. As a consequence, losses in the core can affect the response of the CMC at frequencies where its response is capacitive, thus increasing the complexity of the required circuit model [15], [16]. In recent years, important efforts have been made in the development of techniques for obtaining high-frequency circuit models of CMCs. Generally speaking, these techniques fall into two categories. The first category could be referred to as CMC design methods. This category comprises methods that make use of approximated analytical formulas or electromagnetic simulations to obtain the parameters of the circuit model of the CMC [12], [16]–[19]. These methods are appropriate for CMC design but they require to know details about CMC construction (e.g., number of turns, wire diameter and position of winding on the core) and electromagnetic properties of the core (permeability and permittivity) which might not be available. A second category, within which the method proposed in this work can be included, encompasses methods aimed to obtain the parameters of a CMC circuit model from measurements performed on the CMC [13], [20]–[24]. These techniques are applied to an already available CMC and, for this reason, we will refer to them here as CMC characterization methods. It is worth noting that CMC characterization methods can be complementary to CMC design methods, since they can This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2023.3286007 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on June 15,2023 at 09:27:29 UTC from IEEE Xplore. Restrictions apply. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. -, NO. -, MONTH 2023 2 be used to check the actual circuit parameters or find a more accurate circuit model for a designed and constructed CMC. In this work, we present a CMC characterization method that is supported by a rigorous modal analysis and whose main contributions compared with already reported methods are higher simplicity, accuracy and generality. Specifically: 1) Simplicity of the measurement technique, which does not require many different setups, sophisticated instrumentation, cumbersome calibrations or ancillary circuitry. 2) Simplicity of the circuit model and of the method to determine its parameters. This is due to the fact that the proposed circuit model is organized as separate circuit blocks independently associated with the modal admittances of the CMC and defined attending to the physical effect that they are intended to account for. As a consequence, no post-processing of the parameters is required to ensure physical meaning (e.g., passivity) of the resulting circuit model. 3) Generality and precision of the results. Different types of CMCs, with different core materials and winding structures (stacked or single-layer turns), can be characterized with a circuit model that is typically very accurate up to frequencies close to 100 MHz. The circuit model of the CMC provided by this method does not account for external effects, such as mutual coupling between the CMC and other components of an EMI filter. These effects must be added in a subsequent step of the modeling process for a particular EMI filter. This manuscript is organized as follows: in Section II the natural modes of any circuit model of a symmetric CMC are identified, the contribution of the modal admittances of the CMC to the symmetric and asymmetric responses of the CMC is analyzed, and, based on this modal analysis, a circuit model especially devised to facilitate generality, accuracy and easy determination of its parameters is proposed for the CMC. The proposed characterization technique, based on two simple measurement setups and a decision-making algorithm, is expounded in Section III. Experimental validation of the proposed characterization technique is presented in Section IV and, finally, conclusions are summarized in Section V. II. ANALYSIS A. Natural modes of a CMC Fig.1 shows a four-ports device connected to a singlephase power line. Single-phase power lines and, in general, differential lines can be regarded as (N+ 1)-conductors transmission lines with N= 2. These lines can in general support two transmission modes [25]. In particular, CM and DM are in general analyzed in these type of lines. The main reason is that, from an EMC perspective, this allows separating the contribution of two modes with different mechanism of generation and requiring different techniques of attenuation [9]. This explains also why, in general, CM/DM mode conversion should be avoided in filtering devices, such as CMCs, as well as in transmission lines. Regarding this, it is important TABLE I THE FOUR NATURAL MODES OF A SYMMETRIC FOUR-PORTS NETWORK Name Voltage mode Modal admittance Ground (G) vG= [+1,+1,+1,+1] YG Intra-winding (W) vW= [+1,+1,−1,−1] YW Common (C) vC= [+1,−1,+1,−1] YC Differential (D) vD= [+1,−1,−1,+1] YD Fig. 1. A four-ports device connected in a single-phase or differential transmission line. A port is formed between the (L or N) conductor and the reference conductor at each one of the four terminals of the device. to highlight that CM and DM are natural modes of threeconductor lines that are cyclic-symmetric (i.e., made up of two similar conductors placed symmetrically with respect to a ground conductor) [25]. This symmetry of the line ensures no mode conversion, which could be an issue for electricallylarge lines. In general, CMCs are four-ports devices intended to provide some degree of attenuation to both the CM and DM signals in a single-phase line without causing significant conversion between CM and DM modes. Moreover, due to its symmetric construction, CMCs have input/output symmetry. This means that its input ports (1 and 3 in Fig.1) can be interchanged with its output ports (2 and 4 in Fig.1) without affecting the attenuation that the CMC provides. In Appendix A is rigorously demonstrated that any four-ports network with input/output symmetry that causes no conversion between CM and DM on a three-conductors power line must have the set of four natural modes given in Table I, referred to in this work as G, W, C and D modes. Therefore, the voltage excitations and modal admittances in Table I are respectively the eigenvectors and eigenvalues of the admittance matrix of the four-ports network. This admittance matrix must therefore be diagonal when expressed in the basis of its eigenvectors, and the modal admittances must be the terms in the diagonal of that matrix. Regarding the physical meaning of these four modes, note that the C mode corresponds to an excitation of the CMC with a CM current flowing on the line, whereas the D mode is related to an excitation of the CMC with a DM current. Therefore, both modal admittances, YCand YD, should be inductive at low frequencies. The W mode instead represents an ”inter-winding” excitation of the CMC (that is, a difference in potential imposed between the two windings of the CMC) and therefore its modal admittance YWshould have a capaciThis article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2023.3286007 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on June 15,2023 at 09:27:29 UTC from IEEE Xplore. Restrictions apply. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. -, NO. -, MONTH 2023 3 tive nature. Finally, the G mode represents a net voltage of the four windings with respect to the common reference (ground). Therefore, YGshould also be capacitive and typically quite small. It is worth pointing out that the modes in Table I have been previously identified as the natural modes of a particular circuit model of the CMC in [24]. However, note that the result presented here is much more general, since we have demonstrated that those four modes are always the natural modes of any four-ports circuit model of a CMC. (a) (b) Fig. 2. Setups for characterizing a CMC according to CISPR-17 standard [26] with a spectrum analyzer (SA) featuring a tracking generation (TG). (a) Common mode or asymmetrical mode setup. (b) Differential mode or symmetrical mode setup. B. Insertion loss of a CMC Since CMCs are filtering devices, it is in general of interest to be able to predict and measure the insertion loss offered by a particular CMC for CM and DM noise signals. The setups proposed in the CISPR-17 norm to measure the loss of insertion of four-ports filtering devices are represented in Fig.2 [26]. The transmission coefficients (inverse of the insertion loss) of a CMC in the asymmetric and symmetric mode setups in Fig.2a and Fig.2b can be expressed in terms of the modal admittances of the CMC defined in Table I as follows: Sasym 21 =2R(YC−YG) (2RYC+ 1)(2RYG+ 1) (1) Ssym 21 =2R(YD−YW) (RYD+ 2)(RYW+ 2) (2) These expressions can be obtained from equations (15) and (17) in Appendix A by realizing that for the asymmetric mode setup the normalizing impedance is 2Rand thus yC= 2RYC and yG= 2RYG, whereas for the symmetrical mode setup yW=RYW/2and yD=RYD/2. An important conclusion that can be drawn from (1) and (2) is that for a symmetric four-ports device that does not cause CM/DM conversion (e.g., a CMC) only its C and G natural modes will be excited in the asymmetric setup, whereas only its W and D natural modes will be excited in the symmetric setup. Fig. 3. High-frequency modal-parameters circuit (MPC) model of a CMC proposed in this work. The perfect coupling between the two LC/2inductances and between the four LD/8inductances ensures that the CM block is not excited by either W or D modes and that C mode does not excite the DM block. In addition, the YWadmittance is not excited by D mode, and the inductances LD/8are short-circuited by W mode. Therefore, each parameter of this circuit model is associated with only one natural mode of the fourports network. TABLE II MODAL ADMITTANCES OF THE MPC MODEL OF A CMC SHOWN IN FIG.3 Mode Admittance GYG=jωCg≃0 WYW=1 1 jωCW +1 YLCRW CYC= 2(jωCC+1 jωLC +1 RC +YRLC) DYD= 2(jωCD+1 jωLD +1 RD ) C. Modal-parameters circuit model of a CMC From the discussion in Sections II-A and II-B, it is clear that an accurate circuit model of a CMC should predict the actual insertion loss offered by the CMC in both the symmetrical and asymmetrical setups in Fig.2 within a sufficiently wide frequency range. One of the key ideas of this work is that to obtain a circuit model of a CMC that is at the same time accurate, as simple as possible and applicable to different types of CMCs, this circuit model should be made up of different circuit blocks that independently account for the modal responses of the CMC and whose basic structure is defined with a physical criterion based on the expected response for each modal excitation. Following this criterion, in this work we propose the general circuit model of a CMC shown in Fig.3. The admittances of the natural modes of that circuit model are listed in Table II. The circuit model in Fig.3 is divided into two different blocks. The CM block, accounts for the response of the CMC to its C mode, which is excited by the asymmetric mode setup in Fig.2a. Physically, this response must This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2023.3286007 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on June 15,2023 at 09:27:29 UTC from IEEE Xplore. Restrictions apply. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. -, NO. -, MONTH 2023 4 be inductive at low frequencies and must present a resonant response at high frequencies followed by a capacitive behavior corresponding to the dominance of the electric coupling in the Sasym 21 transmission coefficient of the CMC [24]. Also, the CM block features an optional YRLC parallel admittance that allows realizing a Foster II partial fraction expansion of the admittance of this block [27]. This will allow us to account for two effects: a variable CM inductance associated with a frequency-dependent permeability of the core [13], and also dielectric losses that may be noticeable in some CMCs with high-permittivity cores [16]. Further details will be provided in Subsection II-D. The DM block in Fig.3 accounts for the response of the CMC to both the D and W modes, which are excited by the symmetric mode setup in Fig.2b. However, the elements of the circuit involved in the D and W responses of the CMC are kept separated within the DM block. This is achieved thanks to the perfect mutual coupling between the four LD/8 inductances present in this DM block. It can be checked that this arrangement ensures that the response of the CMC to a W mode is given only by the admittance YW, while the position of this admittance YWjust in the symmetry plane of the DM block causes that YWdoes not affect the modal admittance YD of the CMC. Also, note that the proposed schematic gives rise to a YDwhich is basically that of a parallel RLC tank. This accounts for the effect of the leakage inductance of the CMC (dominant at low frequencies) and the capacitive couplings that should dominate the YDadmittance (and therefore the Ssym 21 of the CMC) at high frequencies [28]1. (a) (b) Fig. 4. Alternative forms of modal admittance YW. (a) Single capacitance. (b) Capacitance in series with a resonant parallel LCR network. Regarding YW, since it models an inter-winding electric coupling, it must physically have a capacitive nature. However, as we will see in section IV, dielectric losses and internal inter-turns resonances, related to the small wavelength inside the core [15], may also have an impact, especially for core materials with high permittivity. For this reason, depending of the response of the particular CMC analyzed, the impedance YWis allowed to be either a single capacitance or a capacitance in series with a parallel LCR network, as illustrated in Fig.4. Finally, with respect to the capacitances Cgin the circuit model in Fig.3, they have been included for generality. However, in this work the effect of this admittance to ground is disregarded (i.e., we consider YG= 0) due to the following general and practical considerations: 1Since electric couplings inside the CMC are distributed effects between turns of each winding and between windings, it is reasonable to assume that the electric field distribution inside a particular CMC might be different in C and D modes. For this reason, the capacitor CDin the DM block is allowed to be different from CCin the CM block in the circuit model in Fig.3. 1) In general, it is reasonable to expect that YGis quite small compared to the rest of the modal admittances of the CMC as long as metallic surfaces such as the return plane of the EMI filter or a possible grounded shielding cage are not quite close to the CMC. Of course, this condition can be easily met when performing the characterization of a stand-alone CMC by avoiding the near presence of metallic surfaces. 2) The YGadmittance depends not only on the CMC but also on the details of its mounting. Therefore, it does not make sense to quantify this (presumably small) effect during the characterization process of a standalone CMC. A better approach should be to assess this effect and, if necessary, incorporate it into the whole model of a concrete filter in a subsequent stage of the design process. 3) The impact of coupling to ground should slightly improve the attenuation of the CMC at high frequencies. Therefore, disregarding YGputs us in a conservative worst-case scenario. Disregarding YGin the CMC model is further justified by the experimental results in section IV-B, which confirm that attenuation of a single-stage EMI filter can be very accurately predicted up to 100 MHz by a circuit model that disregards coupling of the CMC to the return plane. In summary, the proposed circuit model for a CMC includes different circuit blocks, each one easy to identify and independently accounting for the response of only one of the natural modes of the CMC. Because each parameter of this circuit model is involved in the response of the CMC to only one of its natural modes, we will refer to this circuit model as the modal-parameters circuit (MPC) model of the CMC. D. The optional YRLC admittance A key feature of the MPC model is that the basic circuit structures of its blocks correspond to what physically should be expected for the particular mode that they represent. Also, optional additional impedance blocks are proposed within some of these blocks to account for second-order effects that may be noticeable in some CMCs. In particular, the MPC model in Fig.3 includes and optional YRLC admittance within the CM block to account for effects such as a frequencydependent CM inductance or dielectric losses. Regarding the first effect, it has been demonstrated in [13] that CMCs with nanocrystalline cores may exhibit a frequency-dependent CM inductance due to the frequency dependence of the permeability of the material. In that work, this effect is accounted for by adding a parallel admittance made up of one or more series RL circuit branches to the CM block of the circuit model of the CMC proposed in that work. The basic idea is to perform a convenient Foster II partial fraction expansion of the admittance of the CM block [27]. The exact number of required RL branches and the value of its parameters can be easily determined by adding successive RL branches to the original circuit model until a good approximation of the OC response of the CMC is obtained. This idea can be conveniently incorporated into the MPC model. This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2023.3286007 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on June 15,2023 at 09:27:29 UTC from IEEE Xplore. Restrictions apply. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. -, NO. -, MONTH 2023 5 The optional YRLC admittance Fig.3 can also be shaped to account for the effect of dielectric losses, which affects the capacitive response of some CMCs. In particular, CMCs with windings arranged as a few turns wounded in a single layer over a high-permittivity core. In those CMCs, the electric coupling through the core may dominate the overall electric coupling between turns [15]. Therefore, if dielectric losses are noticeable, which occurs for instance for some CMC with MnZn ferrites or nanocrytalline cores, the effect of the electric coupling cannot be accurately modeled by the single CCcapacitor included in the modal YCadmittance in the MPC model in Fig.3. In this work we will show that this effect can be accounted for by performing a Foster II partial fraction expansion of the YCadmittance that consists in adding series capacitor-resistor (CR) branches in parallel with its CCcapacitor. In this way, the modal admittance YCcan be expressed in general as follows: YC= 2jωCC+1 jωLC +1 RC + n X k=0 1 Rk+1 jωCk | {z } YRLC (3) Where Rkand Ckstand for the resistors and capacitors of the successive parallel CR branches that can be added, if necessary, to the YRLC admittance. Regarding YRLC, it is interesting to note that we have experimentally verified that a similar parallel admittance is not typically required in the DM block of the MPC model in Fig.3, which leads to a simpler form of YDcompared with YC. The independence of the leakage inductance of the CMC (LDwithin YD) with frequency can be explained by the fact that the equivalent model of a CMC excited by a DM is that of a single coil with a rod core [28]. On the other hand, CD within YDhas only an effect at very high frequencies. Thus, even though dielectric losses might in principle affect this capacitive response, its impact on the accuracy of the model is not significant. III. CHARACTERIZATION TECHNIQUE This section presents a characterization technique to obtain the parameters of the MPC model of the CMC proposed in Section II. This characterization technique is based on performing an efficient search to determine the parameters of the MPC model of the CMC that allow a good matching between measured and predicted transmission coefficients of the CMC for a pair of suitable measurement setups. Also, a decision-making algorithm has been devised to arrive at the simplest version of the proposed MPC model that is accurate for a given CMC. A. Measurement setups Fig.5 shows the two measurement setups, referred to as OC and WD setups, that are proposed in this work to obtain the parameters of the MPC model of the CMC. The transmission coefficients of the CMC when connected as per those two (a) (b) Fig. 5. Proposed setups for characterizing a CMC. Measures can be taken with a vector network analyzer (VNA) or with a SA featuring a TG. (a) Open circuit setup (OC). (b) Capacitive intra-winding setup (WD). setups can be expressed in terms of its natural admittances as follows: SOC 21 =2RYCYD 2RYCYD+YC+YD (4) SWD 21 =2RYDYW 2RYDYW+YD+YW (5) The expression of SOC 21 in (4) reveals that both the C and D natural modes of the CMC are excited in the OC setup in Fig.5a. Since both YCand YDadmittances correspond to parallel RLC resonators, two resonance peaks are expected in the magnitude of SOC 21 . First, a resonance related to YC(i.e., the CM block in Fig.3) and another resonance at higher frequency related to YD(i.e., the RLC resonator of the DM block in Fig.3). This makes the OC connection especially suitable for characterization of the YCand YDmodal admittances of the CMC. For this reason, the OC connection has already been proposed in [24] for the characterization of CMCs. However, the parameters of the relatively simple circuit model of the CMC proposed in that work are not independently assigned to particular modes. Therefore, the characterization technique proposed in [24] is based on the assumption that the response of the CMC to the W mode, which is not excited by the OC setup, can be inferred from the measurement of SOC 21 . We will show in section IV that for some CMCs this assumption is not valid, thus yielding inaccurate circuit models for the CMC. As an alternative, in this work we propose the WD setup in Fig.5b to characterize the response of the CMC to excitation in the W mode. To understand the interest of this connection, note that SWD 21 in (5) represents the transmission coefficient of the series connection of the YWand YDadmittances, which implies that this transmission coefficient will be determined by the lower of these two admittances. Because YWaccounts for the capacitive coupling between the windings and YDcorresponds to a parallel RLC tank, YW≪YDat low frequencies. Therefore, measuring |SWD 21 |should allow an accurate characterization of the actual capacitive response of the CMC in the W mode. Also, possible high-frequency resonance effects in YW, related to inter-turns resonances, can be readily characterized from the measured |SWD 21 |, as we will see in section IV. An additional critical advantage of the WD setup is its simplicity [29]. In fact, note that this measurement can be carried out with the This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2023.3286007 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on June 15,2023 at 09:27:29 UTC from IEEE Xplore. Restrictions apply. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. -, NO. -, MONTH 2023 6 Fig. 6. Flow chart of the decision algorithm proposed in this work to determine the exact schematic of the circuit model in Fig.3 that fits a given CMC. same basic setup as that required to measure the OC response of the CMC by only performing a change of terminals of the CMC connected to the output and input ports of the SA or VNA. It is worth mentioning that the configurations in Fig.5 have been previously proposed, along with other connections, to characterize CMCs [20]. In this regard, the contribution of this work is to make use of the modal analysis presented here to demonstrate that these two simple configurations are sufficient to characterize any CMC. B. Decision making algorithm The MPC model of the CMC proposed in this work, shown in Fig.3, includes a pair of configurable admittances, namely YRLC and YW, whose exact schematic depends on the particular CMC which is being characterized. This section presents a decision-making algorithm aimed at obtaining, for a given CMC, to a circuit model which is, at the same time, highly accurate and as simple as possible. A flow chart of this algorithm is presented in Fig.6. The basic idea of this algorithm is as follows: the analytical expressions in (4) and (5) allow us to calculate the transmission coefficients SOC 21 and SWD 21 for any set of parameters of the circuit model of the CMC given in Fig.3. To determine the optimal set of parameters that provides the closest match between the measured and the analytical SOC 21 (f)and SWD 21 (f)curves we employ an advanced search algorithm. In this work, we have used a particle swarm optimization (PSO) algorithm [30], but alternatives such as genetic algorithms could be used as well. The PSO algorithm has been programmed to minimize a cost function defined as the sum of the square of the differences between the measured and analytical transmission coefficient curves. Using this basic idea, the decision-making algorithm for determining the parameters of the circuit model of the CMC involves two consecutive search processes, which are represented in Fig.6. Further details of these processes are provided below. 1) OC curve: In a first part, this algorithm makes use of the SOC 21 curve measured according to the OC setup in Fig.5a to determine, by using the parameter search technique described above, the optimal set of elements of the YCand YDmodal admittances that are able to account for this SOC 21 response. Prior to the parameter search process, it is convenient to constrain the search space by obtaining initial values of the parameters to be determined. For example, LCand LDcan be readily estimated by measuring the self-inductance of one of the coils of the CMC when the other coil is respectively opencircuited or short-circuited. Then, the frequencies of resonance of YCand YD, easily identified as the two dips of the |SOC 21 | curve [24], can be used along with LCand LDto estimate the initial values for CCand CD. In this first part of the decision-making algorithm, the number of CR branches included in the YRLC admittance within the CM block, NCR, is initially set to zero. This is valid for CMCs with low dielectric losses. However, if this simple option does not provide an accurate fitting of the curve |SOC 21 |, the algorithm proceeds to successively add parallel RC branches to the YRLC impedance within the CM block of the circuit model until a good agreement between the measured and calculated curves |SOC 21 |is reached. Typically, one or two CR branches are enough to characterize CMCs that exhibit noticeable dielectric losses. As a result of these steps, a set of final values of the parameters of the YCand YDadmittances of the MPC model of the CMC is obtained. Since an analytical expression for SOC 21 (4) is used, this search process is typically quite fast (a few seconds). This first part of the algorithm can be easily upgraded to allow it to also account for the effect that the frequencydependent permeability of the core has on the response of some CMCs. This has been briefly discussed in section II-D. However, since the key aspects of the method have been reported in [13], and its adaptation to the method proposed here is relatively straightforward, these details have been omitted in this section. 2) WD curve: Once the final values of the parameters of the CM and DM blocks of the MPC model in Fig.3 have been determined, the second part of the decision-making algorithm makes use of the measured SWD 21 curve to characterize the YW natural admittance of the CMC. As a first step, YWis assumed to be purely capacitive (YW≈ jωCW). As explained in section III-A, SWD 21 is dominated by YWat low frequencies. Then, a simple value of the measured This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2023.3286007 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on June 15,2023 at 09:27:29 UTC from IEEE Xplore. Restrictions apply. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. -, NO. -, MONTH 2023 7 SWD 21 curve can be used to estimate CW. To determine whether this capacitive YWis a good approximation of the modal admittance of the CMC, the capacitance Csym = 2CD−CW must be calculated. The interest of this parameter lies in the fact that the expression for the symmetric response of the CMC, Ssym 21 (2), is proportional to the difference YD−YW. This is the admittance of a parallel RLC tank whose capacitor, Csym = 2CD−CW, can be either positive (CW<2CD) or negative2(CW>2CD). The first possibility is typical of CMCs with stacked turns and/or low-permittivity core material because in these cases the electric coupling between turns of the same coil is usually higher than the electric coupling between the two windings. Therefore, the response of the CMC in the symmetric setup is the typical resonant response of a parallel RLC tank, resulting in a marked dip in the |Ssym 21 | curve. For these cases, a purely capacitive YWleads in general to a sufficiently accurate circuit model (model A in Fig.6) of the CMC. However, in some cases the high permittivity of the core of the CMC increases the electric coupling between the two windings of the CMC and the condition CW>2CDis met instead, which leads to a negative value of Csym. In this case, no resonant peak must be expected in Ssym 21 . In fact, a smoother response with a shallow minimum is typically measured in the symmetric setup. Moreover, CMCs with Csym <0are prone to additional intra-winding resonances, that can be easily detected in the measured SWD 21 . In these cases, the estimation of the Ssym 21 response of the CMC can be further improved in general considering an enhanced circuit model for the YW admittance that includes a parallel LCR tank in series with the CWcapacitance, as shown in Fig.4b. Therefore, if the calculated Csym is negative, the algorithm assumes that this improved and more complex YWshould be used (model B in Fig.6). The values of the elements of this enhanced YW can be easily calculated using a search algorithm to find a good match between the measured and calculated SWD 21 curves. Several examples of CMCs with Csym <0will be analyzed in section IV and in the supplementary data that accompany this paper [31]. IV. RESULTS To verify the precision and scope of the characterization method proposed in this work, a large number of different CMCs have been measured using a R&S ZND vector network analyzer (VNA). For measurements with the asymmetrical mode setup in Fig.2b, we have used 180odividers constructed with commercial wideband 1:1 transformers (Coilcraft WB2010-1). Since generality of the method is a key aspect to be demonstrated, we have obtained the MPC model for a large number of CMCs with different characteristics. Several representative examples are presented in this section and are listed in Table III. Many other examples are collected in the supplementary documentation provided with this paper [31]. In this section, the results of the characterization of the CMCs obtained with the MPC method proposed in this work 2Note that Csym is a virtual parameter whose sign accounts for the direction of currents flowing at the load in the setup in Fig.2b. The MPC model is a linear, passive and reciprocal circuit with positive capacitances. (a) (b) Fig. 7. SOC 21 and SWD 21 transmission coefficients of the CMC KEMET SU9VR01180. (a) Magnitude. (b) Phase. Parameters of the MPC model are given in Table III and parameters of the SPC model proposed in [24] are given in Table IV. will be compared with the results obtained by using the method previously reported by some of the authors of this paper in [24], which makes use of a simpler circuit model that does not associate its elements with single modal admittances. For the sake of conciseness, in this section we will refer to that circuit model as the simple-parameters circuit (SPC) model, and we will use the SPC acronym for extension to refer to the method proposed in [24]. This comparison between the MPC and the SPC methods is aimed at highlighting the contributions of the MPC method, which, while retaining the relative simplicity of the SPC method, has a much wider scope both in terms of the types of CMCs that it can deal with and in terms of the bandwidth of accuracy of the circuit model obtained. The parameters found for the MPC model for all the CMCs analyzed in this section are listed in Table III, while the parameters found for the SPC model proposed in [24] are given in Table IV. First, it is illustrative to analyze a case for which the two methods work correctly and to analyze what are the necessary conditions for this situation to occur. Fig.7 shows measured and approximated SOC 21 and SWD 21 for the CMC identified as KEMET SU9V-R01180 in Table III. A very good concordance can be observed between the measured transmission coefficients (magnitude and phase) and the approximations provided by both the MPC and the SPC method. Only a slight This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2023.3286007 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on June 15,2023 at 09:27:29 UTC from IEEE Xplore. Restrictions apply. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. -, NO. -, MONTH 2023 8 TABLE III PARAMETERS OF THE MPC MODEL OF THE CMCS CHARACTERIZED IN THIS PAPER Manufacturer Part number L1 (mH) CC (pF) CR in parallel with CCLC (mH) RC (kΩ) CD (pF) LD (µH) RD (kΩ) CW (pF) Csym (pF) LCRWnetwork NCR C1(pF) R1(kΩ) L (µH) C (pF) R (kΩ) KEMET SU9V-R01180 18212.4 0 50.7 245 14.2 153.2 34.2 2.2 26.2 KEMET SC-02-30G 3.323.1 0 7.1 35.3 3.3 7.3 16.1 2.9 3.7 EPCOS B82722A2302N001 1.2 3.3 0 2.7 24.3 3.5 7.6 10.0 16.5 -9.5 9.1 0.3 1.0 WE3 7448261418 1.8 3.7 1 12.2 11.5 3.9 24.9 3.2 9.7 11 16.5 -10.1 8.2 0.3 0.7 1Rated inductance. Information obtained from the data sheet. 2Data sheet provides minimum inductance instead of rated inductance. 3W¨ URTH ELEKTRONIK. TABLE IV PARAMETERS OF THE SPC MODEL (FIG.5 IN [24]) FOR THE CMCS CHARACTERIZED IN THIS PAPER Manufacturer Part number Ct (pF) Cw (pF) LC (mH) RC (kΩ) LD (µH) RD (kΩ) KEMET SU9V-R01180 12.6 0.7 50.2 248 160.8 33.6 KEMET SC-02-30G 3.1 0.2 7.1 35.3 7.3 15.2 EPCOS B82722A230N001 3.3 0.1 2.7 24.3 7.6 10.0 WE 7448261418 17.2 0 3.9 16.9 2.0 6.5 discrepancy can be observed between the |SWD 21 |curve and that provided by the SPC method. Even so, the approximation is not too bad, especially considering that the SPC method does not make use of the |SWD 21 |curve to find the set of parameters of its circuit model and, therefore, SWD 21 is actually predicted from the measured SOC 21 . The modal analysis presented in Section II suggests that a good approximation of the measured SOC 21 and SWD 21 of a CMC should translate into a good prediction of the transmission coefficients of the CMC in the asymmetric and symmetric test setups (i.e., the attenuation provided by the CMC for CM and DM respectively). This is confirmed by the results presented in Fig.8. This figure shows the magnitude and phase of the measured Sasym 21 and Ssym 21 of the KEMET SU9V-R01180 CMC along with the transmission coefficients predicted with both the MPC and the SPC method. Note that the coincidence between measured and predicted values is quite good for both methods. This excellent performance of both methods can be understood by noticing in Table III that the CW(inter-winding) capacitance of this CMC is small compared to CD. Therefore, the impact of the natural admittance YWin the symmetric transmission coefficient is small. Due to this, the improvements offered by the MPC method are not significant in this case. This conclusion holds in general for CMCs with highly stacked winding turns and/or low permittivity cores, since both conditions typically translate into an electric (capacitive) coupling between windings, which (a) (b) Fig. 8. Comparison of measured and predicted Ssym 21 and Sasym 21 transmission coefficients for the CMC KEMET SU9V-R01180. (a) Magnitude. (b) Phase. is small compared with intra-winding capacitance. Another situation commonly encountered is that represented by the CMC analyzed in Fig.9 and Fig.10. For this CMC, Fig.9 shows that whereas both methods provide a good approximation for SOC 21 , the SPC method fails to accurately predict SWD 21 . This results in a poor prediction of the frequency of resonance of Ssym 21 , as shown in Fig.10. To explain this, it is useful to compare the parameters of the MPC model of this CMC (Table III) with those of the CMC previously analyzed (KEMET This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2023.3286007 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: Universidad de Sevilla. Downloaded on June 15,2023 at 09:27:29 UTC from IEEE Xplore. Restrictions apply. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. -, NO. -, MONTH 2023 9 Fig. 9. Magnitude of the SOC 21 and SWD 21 transmission coefficients of the CMC KEMET SC-02-30G. Parameters of the MPC model are given in Table III and parameters of the SPC model proposed in [24] are given in Table IV. Fig. 10. Comparison of measured and predicted magnitudes of the Ssym 21 and Sasym 21 transmission coefficients of the CMC KEMET SC-02-30G. SU9V-R01180). From this comparison it can be observed that while in the previous CMC CWwas quite small compared to CD(CW≪CD), in this case both capacitances are similar. Therefore, YWcannot be disregarded with respect to YDin (5), which implies that an incorrect approximation of YWwill result in a poor prediction of Ssym 21 . Since the SPC method is only based on SOC 21 measurements, which does not excite the YWmodal admittance of the CMC, this admittance is not well approximated by this method, as evidenced by the discrepancy between the measured and predicted |SWD 21 |curves in Fig.9. As a consequence, Ssym 21 cannot be correctly predicted with the SPC method. The improvements provided by the MPC method are even more important for CMCs with negative Csym because this causes a major alteration of the resonant behavior of Ssym 21 , as explained in Section III-B2. An example is presented in Fig.11 and Fig.12, where results are presented for an encapsulated CMC (EPCOS B82722A230N001). It can be seen in Fig.11 that |SOC 21 |is well predicted by both characterization methods. However, unlike the MPC method, the SPC method greatly underestimates CW, which has a relatively large value for this CMC (see Table III). As a consequence, it can be observed in Fig.12 that the prediction of Ssym 21 provided by the SPC method is extremely inaccurate above 10 MHz (differences Fig. 11. Magnitude of the SOC 21 and SWD 21 transmission coefficients of the CMC EPCOS B82722A230N001. Parameters of the MPC model are given in Table III and parameters of the SPC model proposed in [24] are given in Table IV. Fig. 12. Comparison of the measured and predicted magnitudes of the Ssym 21 and Sasym 21 transmission coefficients for the CMC EPCOS B82722A230N001. above 20dB at some frequencies). By contrast, the MPC method provides a good approximation up to approximately 70 MHz with the circuit parameters provided in Table III. From these parameters it can be seen that this CMC exhibits a large YWadmittance with a resonant behavior that makes it necessary to include in the MPC model a series LCRW network such as that shown in Fig.4b to obtain the accurate approximations reported in Fig.11 and Fig.12 for the magnitude of the transmission coefficients. A similar precision is obtained with respect to the phase of these transmission coefficients (not shown for the sake of brevity). An even more complicated situation may arise from CMCs that present both large CWcapacitance and significant dielectric losses in the core. An example is the CMC listed as WE 7448261418 in Table III. Fig.13 shows both the magnitude and phase of SOC 21 and SWD 21 for this CMC. From this graph it can be observed that, unlike in the precedent cases, the SPC method is not able to provide a good approximation of the measured SOC 21 curve. This is due to the fact that the behavior of the curve SOC 21 above its first resonance is not purely capacitive (no 20 dB/dec increase of |SOC 21 | is measured). On the contrary, the MPC method provides a much better approximation of SOC 21 , both in magnitude This article has been accepted for publication in IEEE Transactions on Power Electronics. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2023.3286007 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: Universidad de Sevilla. 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