NIF-Based Frequency-Domain Modeling Method of ... - IEEE Xplore

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 57, NO. 1, FEBRUARY 2015

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NIF-Based Frequency-Domain Modeling Method of Three-Wire Shielded Energy Cables for EMC Simulation Clément Marlier, Arnaud Videt, Member, IEEE, and Nadir Idir, Member, IEEE

Abstract—This paper focuses on the modeling method of energy cables used in power conversion systems, in the aim of EMC simulation and overvoltage analysis. Based on the node-to-node functions method, a simple frequency-domain model with a reduced number of equivalent impedances is considered and applied to three-wire shielded cables, along with a fast identification method based on a cascaded-cell model. Even though the model eventually includes nonphysical virtual impedances, simulation in frequency domain provides accurate results when compared to equivalent experimental measurements, for various cable lengths and in short simulation times. Time-domain waveforms are then extracted from frequencydomain simulation and confirm the effectiveness of the proposed method in a wide frequency range up to 50 MHz. Finally, a good match has been found between experimental and simulation results of voltage overshoots on a buck power converter system. Index Terms—Cascaded-cell model, electromagnetic compatibility (EMC), frequency-domain modeling, power electronics, shielded energy cable.

I. INTRODUCTION OWER electronics converters use power semiconductor devices operating in commutation mode. As a consequence, high voltage and current transients (dv/dt and di/dt) are generated, resulting in conducted and radiated electromagnetic interferences (EMIs) in power conversion systems. These emissions are regulated by electromagnetic compatibility (EMC) standards, which typically focus on frequencies up to 30 MHz in conducted mode (e.g., IEC-61800-3 for variable-speed drive applications) or even more (e.g., DO-160 for embedded systems aboard aircrafts). Conducted emissions are mainly high-frequency commonmode (CM) and differential-mode (DM) currents flowing into the system, and notably in the energy cables that are used for the interconnection between power supplies, loads, and power converters [1]. Depending on applications, these cables can be a few meters to several kilometers long, which exceeds the

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Manuscript received March 1, 2014; revised July 25, 2014; accepted September 10, 2014. Date of publication October 9, 2014; date of current version February 13, 2015. This work has been achieved within the framework of MEDEE (Motors and Electrical Devices for Energy Efficiency), in partnership with Hispano-Suiza. MEDEE is co-financed by European Union with the support of European Regional Development Fund and by the French region Nord-Pas-de-Calais. The authors are with the Laboratory of Electrical Engineering and Power Electronics (L2EP), University of Sciences and Technology of Lille, 59650 Villeneuve d’Ascq, France (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEMC.2014.2359514

wavelengths of the frequencies of interest for EMC standards. Therefore, propagation phenomena must be taken into account, especially because impedances of power supplies, cables, and loads are commonly mismatched in power systems [2], [3]. As a consequence, wave reflection phenomena are likely to occur and cause serious problems such as high-voltage surges across electric motors, reducing the equipment lifetime [4], [5]. EMI issues can be mitigated by improving the EMC filter design [6] or using specific pulse-width-modulation (PWM) strategies, which may spread the noise spectrum using randomized PWM [7], [8] or reduce the CM voltage generated by two-level or multilevel power converters [9], [10]. EMC analysis of a power system requires accurate highfrequency modeling of its components, in the aim of performing either time-domain or frequency-domain simulations. Time-domain simulation is well suited for power converters because semiconductor devices operate in large signal commutation mode, with a strongly nonlinear behavior [11]–[13]. However, frequency domain is best suited for high-frequency modeling, which is required for EMC simulations. Thus, many authors have recently worked on frequency-domain EMC simulation of power conversion systems [14]–[18], even though it is normally not suited for power converters due to their inherent time-varying behavior, which calls for extra techniques to include nonlinearity in frequency domain [19]. Moreover, when interconnection cables have more than two wires or a shield, some authors split the analysis by distinguishing between DM and CM emissions, thereby getting back to two-wire equivalent systems [15], [20]. However, this method does not allow taking into account mode transfer of the conducted emissions between DM and CM, which limits the frequency validity of such models. Multiconductor frequency-domain cable modeling can also be realized using S-parameters [21], [22], which requires prior measurement on the physical device to be modeled. On the other hand, several time-domain modeling methods have been reported with multiconductor and shielded cables [23]–[25]. While frequency-domain models inherently rely on matrix representations, time-domain models require to build equivalent circuits in the form of numerous cascaded cells to account for propagation phenomena in the cables. In this paper, a frequency-domain modeling method is proposed for three-wire shielded energy cables, in the aim of predictive EMC analysis up to 50 MHz, for engineers working on the design stage of a power conversion system. In this regard, two practical constraints are to be considered. First, the final cable to be modeled is not necessarily available for physical

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characterization, which means that the model has to rely only on the knowledge of frequency-dependent per-unit-length parameters; this way, EMC analysis can be performed whatever the cable length. Second, a simulation workflow involving schematic capture of the cable and other system device models (motor, converters, transformer, etc.) is considered, rather than working directly with frequency-domain matrices and transmissionline chain parameters which are more complex in the variablelength multiconductor case [26]–[29]. Thus, the cable model shall be embodied in the form of an equivalent circuit arrangement involving node-to-node virtual impedances [30], and a SPICE-like netlist generated from the schematic is used to perform frequency-domain simulation using an automated modified nodal analysis (MNA) technique [31] in order to build the system equations to be solved at each frequency of interest. The following section first discusses classical cascaded cell models for multiconductor energy cables in time and frequency domains. Then, Section III highlights the practical advantages of frequency-domain modeling for EMC simulation of power conversion systems. As an example, the common configuration of a three-wire shielded cable is analyzed to derive a simple model with a reduced number of virtual unknown impedances. Then, an identification procedure is proposed in Section IV for the model parameterization, making use of equivalent circuit models in order to avoid any measurement on the final cable to be modeled. Finally, experimental results are presented to validate the effectiveness of the proposed method.

II. TIME-DOMAIN MODELING A. Cable Circuit Model in Time Domain Time-domain simulation of a power electronics conversion system requires equivalent electrical circuit models for each component. Cables are usually considered as lossy transmission lines and modeled using cascaded elementary cells of R, L, C, G elements, including coupling factors (k) between conductors, as shown in Fig. 1(a), where the elementary cell is duplicated many times for the whole cable [24]. Depending on the cable length and the targeted maximum frequency of interest, the number of cells is likely to reach several hundreds for EMC simulation of a cable of a few tens of meters. Furthermore, the impedances involved in the elementary cell are frequency dependent, for instance, due to skin effect and proximity effect for the R, L part [32]. In order to take this feature into account in time-domain simulation using fixed impedances, ladder networks are commonly used, as shown in Fig. 1(b) for the R, L part [24]. Similar models can also be used for C, G frequency dependence. Such modeling method has shown to provide satisfactory results for the usual frequencies of interest in EMC simulations. However, the required number of cells and the ladder networks involved in each cell lead to considerable amount of nodes to be solved at each time step, especially when applied to multiconductor cables. Simulation times therefore become an issue. Moreover, convergence issues may appear, which causes problems in time-domain simulations.

Fig. 1. Equivalent circuit model of a three-wire shielded cable (a) elementary “R, L, C, G” cell, (b) example of ladder network approximation for R, L frequency dependency.

Fig. 2. Simple example of three-terminal star-connected device and delta equivalent.

B. Time-Domain Modeling Issues Since the impedances involved in the cable model in Fig. 1(a) have physical meaning, the R, L, C, G values are positive. No- tably, the phases of all complex impedances are in the − π2 , π2 interval (positive real part). However, such a physical electrical model may not be available for a number of electrical devices. Indeed, the high-frequency model of a simple dipole could involve physical effects that may not have simple equivalent electrical model. In such case, an automated procedure like vector fitting is suitable; using a rational approximation of the device impedance, an equivalent circuit may be synthesized with good accuracy [33]–[35]. However, this circuit has no physical meaning and may involve negative values of resistance, inductance, or capacitance. Even though some time-domain simulation softwares allow such nonphysical parameters, stability issues may appear [36] unless passivity of the resulting model is ensured using specific passivity enforcement methods [37]. Such methods apply to multiterminal networks as well, where passivity must be checked for the full network as its internal representation may include nonpassive impedances. For example, the very simple star-connected three-terminal network shown in Fig. 2 has an equivalent delta model in which the ZAB impedance becomes a pure negative resis1 (namely ZAB (ω0 ) = tance at angular frequency ω0 = √2L C − R2LC ∈ R < 0). Even though the complete network remains

MARLIER et al.: NIF-BASED FREQUENCY-DOMAIN MODELING METHOD OF THREE-WIRE SHIELDED ENERGY CABLES

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passive, nonphysical internal current loops are thereby created and the equivalent circuit model may become complex. Considering convergence issues in time-domain simulation of power electronics systems, such problems can be avoided by the frequency-domain approach. III. FREQUENCY-DOMAIN MODELING Since frequency-domain simulation does not require such equivalent circuits, passivity and convergence issues are not relevant, and it performs well in terms of speed. Especially, frequency-domain simulation can be split in a number of frequency ranges to be independently solved and, thereby, benefits from the current trend in processing power, that is, paralleling of computer cores rather than increasing single core speed (unlike single-threaded time-domain simulations). In this regard, it should be noted that simulation time increases with the total number of nodes of the system to be solved. Therefore, building simple models with reduced number of nodes is necessary to allow short simulation times and, thereby, subsequent optimization procedures using these models. This especially applies to energy cables since equivalent circuit models require a large number of nodes, of which most are uninteresting anyway because intermediate currents and potentials are not accessible in the real system. In the following, a multiconductor energy cable is modeled so as to reduce the node count to only the interconnection points with other system devices. Still, the model relies on a circuit description involving equivalent impedances, which makes it suitable for direct inclusion into a schematic capture software. MNA method is then applied to solve the system in frequency domain. As a common test case, a three-wire shielded cable is considered. It is known from circuit theory that an n-terminal system can be represented as a network involving n (n2−1) equivalent impedances, where each pair of terminals is connected through one impedance. Such models have been referred to as “nodeto-node impedance functions” (NIFs) and applied to transformers in [30]. Besides, it is important to notice that the resulting impedances have no physical meaning, so they may not have a physical behavior. We propose a method to apply NIF models to power cables. Since a three-conductor shielded cable can be considered as an eight-terminal system (three phases plus the shield at both ends), it can be modeled as a network consisting of 28 equivalent impedances. Propagation phenomena and frequency dependence of per-unit-length parameters are all taken into account in the obtained model, which includes only a small number of impedances compared with a conventional cascaded-cell model. Furthermore, the cable is a rather simple system when symmetry properties are considered, and its geometry allows the following assumptions. 1) The three conductors can be considered equidistant from each other, as well as from the surrounding shield; thus, the wire-to-wire per-unit-length impedances are the same and so are wire-to-shield impedances. 2) The cable has two ends of four terminals each (three wires plus the shield) and there is an extra symmetry considering that those two ends are seamlessly interchangeable.

Fig. 3.

Full-cable model simplified to six different “impedances.”

TABLE I DESIGNATION OF THE DIFFERENT CABLE VIRTUAL IMPEDANCES FOR THE FREQUENCY-DOMAIN MODEL Symbol

Description

Zw Zw w Zw w c Zs Zw s Zw sc

wire impedance wire-to-wire impedance wire-to-wire cross impedance shield impedance wire-to-shield impedance wire-to-shield cross-impedance

As a consequence, the simplified model of the full cable, including 28 equivalent impedances between its terminals (there are no more cascaded cells), can be represented as shown in Fig. 3. Since these impedances do not actually exist (for instance, there is physically no current flowing directly from one end to the other of the cable with the same instantaneous value), they will be called “virtual impedances.” The six different frequencyvarying complex virtual impedances are reported in Table I. It must be noted that interconductor virtual impedances (wire-towire or wire-to-shield) can be defined either on the same end of the cable or across its two ends: to distinguish between them, the latter is referred to as “cross impedance” in Table I. Evidently, we also define the corresponding complex admittances Yk = Z1k for k ∈ {w, ww, wwc, s, ws, wsc}. This way, the whole cable is restricted to an equivalent circuit model with only six parameters to be identified. Since these impedances have no physical meaning, they are not even passive. As a consequence, a vector-fitting algorithm, even including passivity enforcement, could not synthesize a passive equivalent circuit of each virtual impedance. Still, these impedances are suitable to perform frequency-domain simulation and get a physical result, which can be further postprocessed by inverse Fourier transform if a time-domain result is of any interest. Furthermore, this method does not add any extra node in between the eight cable terminals, which is interesting regarding simulation speed because it avoids potential calculation of unnecessary and even nonphysical nodes. The obtained model is valid for a

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Fig. 4. Direct virtual impedance measurement using an impedance analyzer. (a) Connection setup to the high, low, and guard pins. (b) Equivalent circuit assuming perfect connections.

(so-called “virtual ground”), no voltage appears across the Z 23...26 parallel impedances and the I 0 current is zero. Therefore, the measured current I L is the one flowing into the targeted impedance Z 12 , of which the value is directly given by the anV alyzer I H measurement, even though such impedance does L not physically exist in the real system. For this reason, possi bly nonpassive phases beyond the − π2 , π2 interval may naturally be obtained by the impedance analyzer, as will be shown in Section C. Such direct measurement methods are suitable for compact systems, where terminals are close to each other and cannot be moved such as transformers [39], [40]. Indeed, displacing any terminal would change internal coupling between all terminals (especially at high frequencies), and accordingly disturb the equivalent impedance measurement. Moreover, great caution must be taken regarding the zero-impedance assumption for short-circuiting wires, which must be as short as possible for practical measurements to be reliable in high frequency. In case of a cable, closeness can only be ensured between terminals that are located on the same end of the cable, and the 50-MHz modeling objective is rather constraining regarding the measurement setup. Furthermore, it is a common situation that per-unit-length frequency-dependent cable parameters are known (possibly through characterization on a small-sized cable sample) and that a model is required for different cable lengths. However, direct measurement methods do not allow this because they are based on the final device to be modeled. These are the reasons why such direct measurement methods have been rejected, and an indirect measurement method is proposed in the next section, based on virtual measurements using a prior cascaded-cell model. Precisely, the guarding technique in Fig. 4 will actually be applied by simulation on the prior circuit model. In this regard, obtaining the final cable model can be considered as a model reduction procedure. B. Indirect Identification

given cable length. The next section details the identification procedure for the six impedances. IV. DETERMINATION OF MODEL IMPEDANCES A. Direct Measurement Determination of the six virtual impedances can be done in different ways. For instance, one possibility is to short circuit a number of terminals in order to measure an equivalent impedance between two remaining terminals, and repeat the operation using different cable terminals until enough equations are available to solve an impedance equation system [30]. It should be noted that direct measurement of a single virtual impedance is also possible using an impedance analyzer such as Agilent 4294A with all other terminals connected to the guard. Indeed, Fig. 4(a) proposes a setup example for a six-terminal device, and Fig. 4(b) provides an explanation on how the guarding technique [38] can be used for this. Because the low and guard terminals are kept at the same potential

Once the evolution of the cable per-unit-length parameters with frequency is known, a cascaded-cell model as shown in Fig. 1(a) can be derived. Even though such model is not practical for full system simulation, it provides two interesting features. 1) Creating a model for a cable of a given length is as easy as cascading the appropriate number of cells to achieve the desired length. 2) A circuit simulation software can easily perform ac analysis to determine impedances in various conditions (e.g, short-circuiting some terminals). Thus, the knowledge of per-unit-length parameters is sufficient to derive the reduced 28-element model for a cable of any length, as shown in the following procedure. First, a small-size cable sample is used to measure per-unitlength parameters according to frequency in different configurations (typically, DM and CM, with cable output in short circuit or open circuit). For that means, an identification procedure as described in [24] has been applied. Then, frequency dependence of R, L, C, G parameters is modeled on the whole frequency


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Fig. 6. Example of direct virtual measurement of the Z w s impedance for the reduced model.

Fig. 5.

Cable circuit model of one elementary cell.

range using ladder networks. In this paper, a three-wire shielded cable dedicated to aerospace applications has been used. Considering a maximum interest frequency above 50 MHz and a propagation speed of about half the light speed in vacuum, the smallest wavelength of interest λ does not exceed a few meters. Therefore, the elementary cell has been chosen to represent λ requirement. The 10 cm of cable, which is below the typical 10 corresponding equivalent circuit has been determined with the values presented in Fig. 5. It should be noted that the measured relative permittivity of the dielectric used in this cable is constant over the whole frequency range, which is the reason why ladder networks are not required for the C, G elements. Finally, as many such elementary cells as required are cascaded in order to achieve the desired cable length. For instance, a 17-m sample of the cable has been modeled using 170 elementary cells; in this case, the full-cable circuit model includes more than 4000 nodes to be solved at each time step when using a time-domain simulation software. Based on the full-cable circuit model, reduction to the eightterminal circuit shown in Fig. 3 can be done by reproducing the direct measurement method from Fig. 4, as a virtual measurement in the simulation software. Given two terminals between which a virtual NIF impedance is to be determined, the procedure includes the following steps. 1) An ac voltage VH is applied across the two terminals. For instance, in Fig. 6, the high side H of the ac signal is connected to terminal W1 and the low side L is connected to terminal S. 2) All other terminals are short circuited and, then, connected to the low side of the ac signal, creating a new node G (which has the same potential as L), while leaving room for measuring the current IL between L and G. 3) AC analysis is performed by sweeping over the frequency range of interest, and the frequency response of the IL current is collected. 4) The requested virtual impedance is obtained as the ratio of the complex ac voltage V H by the measured current I L .

Thus, Zws can be determined as a function of frequency in the example of Fig. 6. All six impedances from Table I can be likewise determined using one virtual measurement per impedance. Such indirect identification makes it possible to build the frequency-domain cable model in a predictive way. Indeed, only the per-unit-length parameters of the cable are required (a small cable sample is enough for experimental characterization). Therefore, performing comparative simulations of a power system using several cable lengths is not difficult using the proposed method. C. Actual Determination of Cable Parameters The method described in Section B has been applied on two different portions of the same initial cable: a 5-m-long and a 17-m-long sample. Prior to computing virtual impedances, the elementary cell shown in Fig. 5 has been determined and used in PSpice software to build full-circuit models of the two cables, including respectively 50 and 170 elementary cells. Then, indirect identification as presented in the previous section has been performed for both lengths. The resulting complex parameters for the 5-m cable are shown in Fig. 7. Both magnitude and phase are presented according to frequency ranging between 150 kHz and 50 MHz. Since phases are defined modulo 2π, phase plots are usually stuck within the [−π, π] interval (or even − π2 , π2 for physical impedances due to positive real part). However, in this case, the nonpassivity of virtual impedances allows negative real parts and phase may even turn past the ±π limits. Forcing the 2π modulo would be mathematically correct but would induce many phase discontinuities from π to −π and make the plot less readable. This is the reason why in Fig. 7, phase values are let free to remain continuous and, thereby, reach several times 2π (only the first point at 150 kHz has been arbitrarily forced within [−π, π]). This behavior should be related with propagation phenomena that are implicitly included in these virtual impedances, generating more and more delay as the frequency increases. Furthermore, a similar interpretation can be driven with the cable length, as shown in Fig. 8 for the 17-m cable. Indeed, longer propagation time shifts the phase rise toward lower frequencies, and the

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Fig. 7. Calculated frequency-domain impedances of the 5-m-long cable. (a) Wire and interwire impedances and cross impedances. (b) Shield and wire-toshield impedances and cross-impedances.

Fig. 8. Calculated frequency-domain impedances of the 17-m-long cable. (a) Wire and interwire impedances and cross impedances. (b) Shield and wireto-shield impedances and cross-impedances.

phase shift increase is much larger up to 50 MHz. The next step consists in validating the proposed reduced model of the cable.

B. Model Validation

V. EXPERIMENTAL VALIDATION A. Cable-Only Impedance As a first validation step, physical impedance measurements in short- and open-circuit configurations have been carried out on the 17-m-long cable (using HP 4294 A impedance analyzer) and compared with the model response. The setup, shown in Fig. 9(a), has been chosen to be different from the measurement configurations that were used for the characterization of perunit-length parameters. The result in Fig. 9(b) reproduces typical impedance variations according to frequency, where the characteristic impedance can be determined around 52 Ω. This result confirms that the obtained model using the virtual impedances from Fig. 8 is in good agreement with the measurement up to 50 MHz.

In order to validate the model, the cable has been inserted into a linear system consisting of a function generator and a load, as presented in Fig. 10. The function generator outputs a 90-V square-wave voltage (at maximum power), at 20 kHz (which is a common switching frequency for power converters), and 20% duty cycle. The load has been designed to be fed by a power converter; it is composed of a series connection of resistors and an inductor, placed in an aluminum housing that is connected to the ground in order to stabilize its CM impedances (see Fig. 15). Therefore, the load has three terminals and can be modeled as three Δ-connected virtual impedances that are measured directly by an impedance analyzer according to the procedure presented in Fig. 4. It must be noted that finding an equivalent circuit arrangement “by hand” would be quite difficult to represent this load up to 50 MHz, which in particular involves distributed couplings to ground over the length of all devices, as well as modeling difficulties for the inductor. Similar


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Fig. 11. Frequency-domain simulation results compared with experimental measurements for the 5-m cable. (a) Output voltage v o . (b) Output ground current ig . Fig. 9. Differential-mode impedance of the 17-m-long cable in short-circuit and open-circuit configurations. (a) One-wire to one-wire measurement configuration. (b) Comparison of measurement versus simulation results.

Fig. 10.

Experimental configuration including the power source and the load.

to the cable model, the three Δ-connected virtual impedances (named Zdm , Zcm 1 , and Zcm 2 in Fig. 10) have no physical meaning, which has no adverse effect for frequency-domain simulation, and Fig. 4 setup is therefore convenient to build an accurate model directly from measurements. Finally, the cable is used in a nonsymmetric manner (having two wires that are short circuited at both ends), which intentionally provokes mode transfers in order to test the model robustness. The shield is connected to the load ground on one side and to the source reference potential on the other side. Four quantities have been acquired by a 12-bit oscilloscope: input voltage and current (vi and ii in Fig. 10), output voltage vo , and output ground current ig , using 100-MHz-bandwidth

differential voltage probes and 200-MHz-bandwidth current probes. FFT has then been performed to compare with frequency-domain simulation results. Figs. 11 and 12 present magnitude results for the 5-m and 17-m cable, respectively (for the sake of space, only the output quantities vo and ig are represented, though the results for vi and ii are very similar). In addition to the raw spectra, the upper envelopes are plotted in bold line to allow meaningful comparison of the results. These figures reveal that a good match is obtained between simulation and experiments over the whole frequency range. Moreover, the deviation that can be seen on the vo spectrum above 30 MHz can be explained by the oscilloscope quantization noise, when the magnitude decreases too much compared with its highest value (even though the 12-bit ADC greatly helps reducing this noise limit compared with conventional 8-bit ones). By applying inverse FFT to the frequency-domain simulation results, time-domain waveforms can be obtained and directly compared with oscilloscope measurements. Fig. 13 shows the results for the 17-m cable. It can be seen that the frequencydomain model reproduces the experimental measurements with good accuracy and that propagation phenomena are satisfactorily represented, as shown by propagation times between input and output quantities and by the wave reflections that are responsible for the several “steps” composing the input and output voltages waveforms. It can further be noted that the ig oscillation

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Fig. 14.

Experimental configuration including a buck power converter.

Fig. 15.

Experimental bench.

around 40 MHz appears on each figure; therefore, this resonance is due to the load itself rather than the cable. Fig. 12. Frequency-domain simulation results compared with experimental measurements for the 17-m cable. (a) Output voltage v o . (b) Output ground current ig .

Fig. 13.

Waveform comparisons in time domain for the 17-m cable.

C. Application to a Power Conversion System In order to reflect practical use in a real-power electronics conversion system, the function generator has been replaced by a buck power converter fed by a 500-V dc voltage source through a line impedance stabilization network (LISN), as shown in Figs. 14 and 15. The converter operates at a 16-kHz switching frequency with a 30% duty cycle. The 17-m-long cable is used so as to provoke voltage overshoots across the load terminals (vo voltage), especially during the fast turn-on commutation of the IGBT. Moreover, a 1-m-long input cable of the same kind is used to connect the LISN to the converter and has been modeled as well using the whole method presented in this paper. This real-case application raises the problem of frequencydomain simulation of a strongly nonlinear system due to switchmode power electronics. In order to cope with the issue of timevarying circuit topology, the multitopology equivalent sources (MTES) method [19] has been applied, so the commutation cell is actually represented by two different topologies of equivalent noise source generators (one of them is presented in Fig. 16) and their individual results are combined through convolution operation involving validity functions of each topology. Thus, the final result takes the system nonlinearity into account, while still being a frequency-domain simulation. The advantage of this method is to allow fast simulations using the frequency-domain NIF model, with accurate representation of the propagation path after the commutation. However, the method is not equivalent to time-domain simulation using nonlinear semiconductor model, so the result get closer to reality while still being an approximation.


Fig. 16.

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Frequency model of the buck converter feeding the load through a long cable.

TABLE II MODEL PARAMETERS OF THE POWER CONVERSION SYSTEM Parameter

Cb u s

Resr

Lesl

L1

L2

L3

L4

C1

C2

C3

Rlisn

Clisn

Llisn

Value

10 μF

9.3 mΩ

21 nH

18 nH

23 nH

58 nH

7 nH

15 pF

27 pF

40 pF

50 Ω

0.1 μF

5 μH

The waveforms presented in Fig. 17 show that the maximum voltage surge around 1000 V is correctly represented on the first overshoot. Subsequent oscillations are also similar in simulation compared with the experiment, even though some deviation can be observed on the wave reflection frequency and damping (a few percents). Since the cable model has been validated in Section B, the differences may come from approximations of the MTES method, accuracy of the converter PCB characterization, or from the ground plane shown in Fig. 15, which has not been considered in the model (see Fig. 16). Still, Fig. 17 shows that this method can be used to estimate voltage surges across the load with good accuracy, and with simulation times reduced from 30 min in time domain to only 3 min in frequency domain, using only one CPU core, which means that this duration could be further reduced using computer paralleling techniques.

VI. CONCLUSION

Fig. 17. Waveform comparisons of voltage overshoots due to power converter switching.

The equivalent series resistance and inductance of the dc-bus capacitor have been measured with the impedance analyzer. The parasitic inductances of the PCB tracks (L1 to L4 ) have been characterized likewise, as well as the capacitances to ground around the switching cell (C1 to C3 ). The LISN is composed of two DO-160-compliant NNBM 8126-A units. The circuit parameters used for simulation are reported in Table II. The impedances associated with the MTES equivalent generators are typically the on-state dynamic resistance and the off-state interelectrode capacitances of the devices, which may be obtained from datasheet or experimental characterization [41].

This paper presents a frequency-domain modeling method of energy cables for EMC analysis, which has been applied to a three-wire shielded cable. Making use of cable symmetry properties, an equivalent-circuit cascaded-cell model is reduced from several thousands to eight nodes involving six different virtual impedances to be determined. After the frequency-dependent per-unit-length cable parameters are characterized, a fast identification method is proposed in order to build the model for a cable of any length. This model allows us a significant reduction of computation times compared with conventional time-domain simulation, which could allow optimization procedures at the design stage of a power conversion system. The validity of the proposed method has been confirmed up to 50 MHz by impedance measurements, experimental tests in a linear system including a power load, as well as overvoltage measurements on a simple power conversion system. Future work may focus on EMC modeling of a more complex system including a

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Clément Marlier received the Eng. degree from Ecole Centrale de Lille, Villeneuve-d’Ascq, France, in 2010, and the Ph.D. degree in electrical engineering from the Laboratory of Electrical Engineering and Power Electronics, University of Lille 1, Villeneuved’Ascq, in 2013. His current research interest includes electromagnetic compatibility of the power converters.

Arnaud Videt (M’10) received the Ph.D degree in electrical engineering from Ecole Centrale de Lille, Villeneuve-d’Ascq, France, in 2008. Since 2008 to 2010, he has been an R&D Power Electronics Engineer with Schneider Toshiba Inverter in Pacy-sur-Eure, France, where his research interests include multilevel inverter control, electromagnetic compatibility, and input power quality for motor drive applications. Since 2010, he has been working as an Associate Professor at the Laboratory of Electrical Engineering, Lille, France. His current research interests include power quality, electromagnetic compatibility, and wide-bandgap semiconductor devices for power conversion

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Nadir Idir (M’93) received the Ph.D. degree in electrical engineering from the University of Lille 1, Villeneuve-d’Ascq, France, in 1993. He is currently a Full Professor with IUT A of the University of Lille 1, where he teaches power electronics and electromagnetic compatibility. Since 1993, he has been with the Laboratory of Electrical Engineering and Power Electronics, University of Lille 1. His research interests include design methodologies for HF switching converters, power devices (SiC and GaN), electromagnetic compatibility of the static converters, HF modeling of the passive components, and EMI filter design methodologies.