Wenhua Tan
email: wenhua.tan@ec-lille.fr

Carlos Cuellar
email: carlos.cuellar@univ-lille1.fr

Nadir Xavier Margueron
email: xavier.margueron@ec-lille.fr

N Idir

A High Frequency Equivalent

Keywords: Common mode choke, electromagnetic interference filter, equivalent circuit, rational function approximation

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I. INTRODUCTION

LECTROMAGNETIC interference (EMI) filters are commonly used solutions for mitigating the conducted emissions produced by power converters [START_REF] Mainali | Conducted EMI Mitigation Techniques for Switch-Mode Power Converters: A Survey[END_REF]. A typical topology of EMI filter with ideal components is shown in Fig. 1. However, a real EMI filter is much more complicated and its performances depend on many factors such as filter topology, magnetic material, grounding method, stray elements of components, and parasitic coupling between components etc. [START_REF] Maillet | High-Density EMI Filter Design for DC-Fed Motor Drives[END_REF]- [START_REF] Wang | Effects of Parasitic Parameters on EMI Filter Performance[END_REF]. Complete physic-based models of an EMI filter or their components require extensive electromagnetic analysis, so they are very complex and specific to limited applications [START_REF] Kovacic | Analytical Wideband Model of a Common-Mode Choke[END_REF], [START_REF] Wu | Design, Modeling, and Improvement of Integrated EMI Filter With Flexible Multilayer Foils[END_REF]. Modeling methods based on measurements are more general, direct and accurate for describing the characteristics of the device under study.

The insertion loss (IL) of an EMI filter is usually measured with 50Ω/50Ω convention. However, the IL of the filter under working condition depends on the impedances of the source and the load, which vary with frequency [START_REF] Tarateeraseth | Accurate Extraction of Noise Source Impedance of an SMPS Under Operating Conditions[END_REF], [START_REF] Kong | Common Mode EMI Noise Characteristics of Low-Power AC-DC Converters[END_REF]. In order to obtain the IL of the filter, many solutions have been reported. In [START_REF] Kostov | Insertion loss in terms of four-port network parameters[END_REF], a four-port measurement method using vector network analyzer is presented. This method consists in a black-box modeling of the whole filter with mixed-mode S-parameters. By post-processing the obtained S-parameter data, the IL of the filter can hence be calculated with any source and load impedances. In [START_REF] Sánchez | A Modal Model of Common-Mode Chokes for Conducted Interference Prediction[END_REF], a modal model of common mode (CM) chokes based on four-port S-parameter measurements is presented. Though modal models enable to analyze the conversion of the noise between differential mode (DM) and CM, they also require compatible modal model of noise sources for simulations, which complicates the modeling process. An alternative is to use equivalent circuit models, which are physic based and are compatible with most of the simulation tools. In [START_REF] Wang | Characterization and Parasitic Extraction of EMI Filters Using Scattering Parameters[END_REF], the equivalent circuit of a DM EMI filter is identified by S-parameters measurements. This approach can correctly extract the parasitic couplings in the filter, resulting in good modeling precision. Impedance measurement is a more frequently used technique which has long been studied for identifying the equivalent circuits of passive magnetic components [START_REF] Margueron | Identifying the Magnetic Part of the Equivalent Circuit of n-Winding Transformers[END_REF], [START_REF] Schellmanns | Multiwinding transformers: a successive refinement method to characterize a general equivalent circuit[END_REF]. Recently, a lumpedelement high frequency (HF) model for CM chokes has been proposed [START_REF] Kotny | High frequency modeling method of EMI filters[END_REF], [START_REF] Kotny | High Frequency Model of the coupled inductors used in EMI Filters[END_REF]. This model can be easily built by extracting the parameters from impedance measurement results and can effectively describe the HF characteristics of the studied CM choke and EMI filter. However, the topology To simplify the extraction process, numerical methods can be applied. In [START_REF] Skibin | Behavioral circuit modeling of chokes with multi-resonances using genetic algorithm[END_REF], a genetic algorithm is used to extract the behavior model of chokes, yielding an accurate and reliable broadband equivalent circuit. Nevertheless, the convergence time of such algorithm is usually long when the problem has a large search space. In [START_REF] Tan | Automatic Identification of Magnetic Component Equivalent Circuits Using Impedance Measurements[END_REF], a physics-based equivalent circuit of CM chokes is built with a rational function approximation (RFA) method based on linear optimization [START_REF] Elzinga | Pole-residue formulation for transient simulation of high-frequency interconnects using householder LS curve-fitting techniques[END_REF]. However, the topology of the equivalent circuit is predefined so prior knowledge about the component is needed.

In this study, a HF model of CM chokes is proposed together with a systematic extraction procedure. This procedure is aided by the RFA method incorporating an iterative scheme (referred as IRFA) and it allows a fast and accurate extraction of the parameters in the model. This paper is organized as follows. In Section II, the HF model introduced in [START_REF] Kotny | High Frequency Model of the coupled inductors used in EMI Filters[END_REF] is reviewed. Next, the RFA and the proposed IRFA methods are presented in Section III. In Section IV, the proposed HF model of CM chokes and its extraction procedure are introduced and then applied to a planar CM choke. Finally, experimental verifications of the procedure as well as some discussion of the results are given in Section V. Section VI concludes this paper.

II. HF MODEL OF CM CHOKES

A. Review of the Existing HF Model for CM Chokes

The equivalent circuit for toroidal CM chokes proposed in [START_REF] Kotny | High Frequency Model of the coupled inductors used in EMI Filters[END_REF] [see Fig. 2(a)] is illustrated in Fig. 2(b). As seen, it is a symmetrical structure and the transformation ratio η is assumed to be unity. This model can describe the behavior of many types of CM chokes as long as the winding symmetry is satisfied. The leakage impedance Z 1 includes the winding resistance, the leakage inductance and their variations due to skin and proximity effects of the winding conductor [see Fig. 2(c)]. The magnetizing impedance Z 2 describes the behavior of magnetic core, including the frequency-dependent inductance and losses [see Fig. 2(d)]. In addition, C e stands for the self-parasitic capacitance of each winding whereas the sum C s +C p +C c is related to the inter-winding capacitances of the component. Examining the admittance Y 2 = (Z 2 ) -1 , it is found that Y 2 contains: 1) One pole at origin: the branch of L 2 ; 2) One stable real pole: the branch of R 2 , R C2 , C 2 and 2 R ¢ ; 3) One pure imaginary pole-pair: the branch of 2 L ¢ and 2 C ¢ . With these elements, the HF behavior of the magnetizing impedance Z 2 is accurately modeled.

B. Method of Parameter Extraction

To identify the parameters of the model shown in Fig. 2(b), five selected impedance measurements are carried out, as presented in Fig. 3. All the parameters of the model are obtained through these impedance measurements, as summarized below: 1) T 0 , T 1 , T 2 and T 3 : parasitic capacitances C c , C e , C p and C s ; 2) T 0 and T 1 : leakage impedance Z 1 ; 3) T 2 and T 4 : magnetizing impedance Z 2 .

The extraction procedure is based on observations of the impedance curves with some typical frequency responses, e.g., 20 dB/Dec as inductance and -20 dB/Dec as capacitance. The values of the parasitic capacitances are calculated using the resonance frequencies. To fit the model with the measurement results, some manual adjustments of the parameters are needed for Z 1 and Z 2 .

Though good accuracy is achieved in [START_REF] Kotny | High Frequency Model of the coupled inductors used in EMI Filters[END_REF], the HF model shown in Fig. 2(b) can be improved for the following reasons:

1) The extraction procedure requires manual adjustments on the parameters of the model to achieve a desired accuracy. As a result, the procedure is complex and time-consuming. Therefore, a computer-aided procedure is preferable.

2) The topology of the equivalent circuit is determined according to experimental observations. For example, the admittance Y 2 =1/Z 2 is chosen to have one stable pole and one imaginary pole-pair from experience. In consequence, a systematic equivalent circuit synthesis method is needed to choose the topology.

In view of these necessities, a HF equivalent circuit model for CM chokes and an IRFA based extraction procedure are proposed to facilitate the process. The IRFA method is first presented in the next section.

III. PARAMETER EXTRACTION USING ITERATIVE RATIONAL FUNCTION APPROXIMATION

A. Brief Review of Rational Function Approximation Method

The impedance Z(s) (or admittance Y(s)) of a linear circuit network can be represented by a rational function.

1 1 1 0 1 1 1 ( ) ( ) ( ) 1 m m m m n n n n b s b s b s b N s Z s D s a s a s a s - - - - + + + + = = + + + +   (1)
with |m-n| ≤ 1 and s = jω. Without loss of generality, only the impedance Z(s) is considered in this section. The same analysis is valid for the admittance Y(s). The goal of the RFA is to search the values of a i and b i to minimize the error between the rational function Z(s) and measured data Z meas (s), as expressed by:

( ) 2 , ( ) argmin ( ) ( ) ( ) i i k meas k k meas k a b k k N s Z s Z s Z s D s -  = å . ( 2
)
where s k = jω k . To handle this nonlinear optimization problem, Levy used a linearization technique [START_REF] Levy | Complex-curve fitting[END_REF]. Stemming from Levy's method, the authors introduced in [START_REF] Elzinga | Pole-residue formulation for transient simulation of high-frequency interconnects using householder LS curve-fitting techniques[END_REF] the RFA method for generating the macro-model of HF interconnects. Reformulating (2) by multiplying D(s k ) gives:

( ) ( ) ( ) 0 k m e a s k k N s Z s D s - = .
(

Separating the real parts and imaginary parts of (3) yields a linear system of a i and b i :

Re ( ) Re ( ) ( ) 0 Im ( ) Im ( ) ( ) 0 k m e a s k k k m e a s k k N s Z s D s N s Z s D s ì é ù é ù - = ï ë û ë û ï í é ù é ù ï - = ï ë û ë û î . ( 4
)
This system is over-determined due to the large number of measured points. In order to calculate the unknowns a i and b i , a least-square method using QR factorization is applied to solve the over-determined system [START_REF] Wang | Effects of Parasitic Parameters on EMI Filter Performance[END_REF]. Once the values of b i are known, the poles of (1) can be derived by solving the roots of the denominator D(s). The impedance Z(s) can then be written into a pole-residue form:

1 ( ) n i i i r Z s d e s s p = = + ⋅ + - å . ( 5
)
It should be noted that r i and p i can be complex-valued. Again, the terms d, e, and r i are solved from this linear overdetermined system (5) by the least-square method [START_REF] Elzinga | Pole-residue formulation for transient simulation of high-frequency interconnects using householder LS curve-fitting techniques[END_REF].

B. Iterative Rational Function Approximation Method

The RFA method is proven to be efficient with noise-free simulated data. However it cannot be directly employed for our application, since Levy's linearization technique is known to have a frequency bias problem [START_REF] Sanathanan | Transfer function synthesis as a ratio of two complex polynomials[END_REF]. In (2), Z meas (s k ) is multiplied by D(s k ) that increases rapidly with s k (as frequency increases). This means the errors of (2) are biased by D(s k ), making HF errors more important than low frequency (LF) errors. As a result, this frequency bias due to D(s k ) causes poor fitting precision at LF. However, to correctly identify elements like DC winding resistance of CM chokes, this lack of fitting precision at LF is undesirable. To handle the frequency bias, the RFA method is improved in this work using the Sanathanan-Koerner (SK) iteration [START_REF] Levy | Complex-curve fitting[END_REF]. The denominator of the iteration t-1 D [t-1] (s k ) is divided on both sides of (3) for the iteration t, giving

1 1 ( ) ( ) ( ) 0 ( ) ( ) t t k m e a s k k t t k k N s Z s D s D s D s é ù é ù ë û ë û é ù é ù - - ë û ë û - = . (6)
It can be seen that the bias due to D [t] (s k ) is alleviated by dividing D [t-1] (s k ), resulting in better fitting accuracy at LF. Based on ( 4) and ( 6), the proposed IRFA formulation used in this work is given by

Re Re Im Im Re ( ) ( ) Re ( ) ( ) ( ) 0 Im ( ) ( ) Im ( ) ( ) ( ) 0
t t t t k k k m e a s k k t t t t k k k m e a s k k P s N s P s Z s D s P s N s P s Z s D s é ù é ù é ù é ù ë û ë û ë û ë û é ù é ù é ù é ù ë û ë û ë û ë û ì é ù é ù ï - = ï ê ú ê ú ï ï ë û ë û í é ù é ù ï - = ï ê ú ê ú ï ï î ë û ë û . (7)
with the weighting factors Re ( )

t k P s é ù ë û
and Im ( )

t k P s é ù ë û
given by

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) * 1 Re 1 1 1 1 * 1 Im 1 1 1 1 ( ) Re Re Im Im ( ) Re Im Im Re t k t k t t t t k k k k t k t k t t t t k k k k D s P s D s N s D s N s D s P s D s N s D s N s é ù - ë û é ù ë û é ù é ù é ù é ù - - - - ë û ë û ë û ë û é ù - ë û é ù ë û é ù é ù é ù é ù - - - - ë û ë û ë û ë û é ù ê ú ë û = é ù é ù é ù é ù + ê ú ê ú ê ú ê ú ë û ë û ë û ë û é ù ê ú ë û = é ù é ù é ù - ê ú ê ú ê ú ë û ë û ë û ì ï ï ï ï ï ï ï ï ï í ï ï ï ï ï ï é ù ï ï ê ú ï î ë û . ( 8
)
It should be mentioned that in Re ( )

t k P s é ù ë û
and Im ( )

t k P s é ù ë û
, a balanced weighting is used to achieve comparable fitting precision for both real parts and imaginary parts. The derivation of ( 8) is detailed in Appendix. The algorithm of IRFA is given in Fig. 4, where the ε iter is defined as the maximum error

( ) ( ) ( ) { } iter max / fit k meas k meas k k Z j Z j Z j e w w w = - . (9)
The RFA method is first performed to provide a starting point, the IRFA is then applied until the stopping condition(s) (in this work: ε max =10%, N max =15) are fulfilled. To show the efficiency of the proposed IRFA algorithm, different methods are applied to fit the measured impedance (from 40 Hz to 70 MHz) of an 8-turn planar inductor with Ferroxcube 3F3-Planar E38 ferrite core. A rational function like [START_REF] Mainali | Conducted EMI Mitigation Techniques for Switch-Mode Power Converters: A Survey[END_REF] with m = n = 6 is used for the fitting. It can be seen on Fig. 5(a) that the RFA fitting results are wrong on LF due to the frequency bias problem and the errors decrease when frequency increases. In . However, the error may exceed the specified error level ε max on the two extremities of fitting range and the resonance frequencies. To achieve a better precision, one can increase the order of the rational function at the expense of obtaining a more bulky equivalent circuit. In order to obtain a simple equivalent circuit, the choice of the order turns out to be a trade-off between the accuracy and the complexity of the circuit.

C. Equivalent Circuit Synthesis

Based on the results obtained by the IRFA method, a systematic approach is used for synthesizing the equivalent circuit [START_REF] Antonini | Spice equivalent circuits of frequency domain responses[END_REF]. Representing (5) by the Foster expansion:

2 1 1 real poles complex conjugate pole pairs ( ) r c N N i i i i i i i i r a s b Z s d e s s p s ms n = = ae ö + ÷ ç ÷ ç = + ⋅ + + ÷ ç ÷ ç ÷ - + + è ø å å    . ( 10
)
Note that the a i , b i , m i , and n i are different variables from those used in [START_REF] Mainali | Conducted EMI Mitigation Techniques for Switch-Mode Power Converters: A Survey[END_REF]. As seen, the Z(s) can be expressed by the sum of a constant term d, an s-proportional term e, real poleresidue terms and complex conjugate pole-residue pair terms. The same expression holds for the admittance Y(s). The constant d corresponds to a resistor and the e•s term corresponds to a capacitor for Y(s) or an inductor for Z(s), respectively. In Table I, the detailed equivalent circuit synthesis methods for real-pole terms and complex-pole pair terms are summarized.

1) Real pole: With a positive residue r i , the equivalent circuit is simply a R-L in series for Y(s) or R-C in parallel for Z(s), as shown in Table I. However, negative r i may appear in the IRFA results, resulting in negative valued elements. Strictly speaking, negative elements are not feasible in real circuit design so that they should be avoided. However, for EMI modeling and simulation purposes, these elements can be accepted for synthesizing an accurate equivalent circuit. Nonetheless, the negative elements are difficult to handle during time-domain simulations. To circumvent this difficulty, we suggest another topology: a positive-valued cell together with a negative resistance, as illustrated in Table I. Suppose that a negative residue r i is extracted with a stable real pole p i (p i <0) for the admittance Y(s). Normally, a negative R-L series cell is obtained. However, according to Table I, the circuit composed of a positive R-C series cell in parallel with a negative resistance -R can be used, where the expression for R, C and -R are given by ( )

/ positive 1 / positive / negative i i i i i R p r C pR R p r ì ï = ï ï ï = - í ï ï ï-= - ï î . ( 11
)
This circuit is equivalent to the negative R-L series cell since they share the same rational function. The negative resistance -R are subsequently merged into the constant term d in [START_REF] Sánchez | A Modal Model of Common-Mode Chokes for Conducted Interference Prediction[END_REF], resulting in a new constant term d * = d-1/R = d-r i /p i . Similar analysis can be done for the case of impedance Z(s). With these circuits, negative elements for real pole terms can be reduced to minimum.

2) Complex-pole pair: To treat complex-pole pair, a fourelement circuit can be used [START_REF] Antonini | Spice equivalent circuits of frequency domain responses[END_REF]. It is the minimal type realization since there are four free variables a i , b i , m i and n i in the rational function (see Table I). Another equivalent circuit referred as "Extended" type, uses a six-element circuit but with easier parameter calculation formulas [START_REF] Ye | Passive-assured rational function approach for compact modeling of on-chip passive components[END_REF]. In Fig. 6, the equivalent circuit corresponding to the admittance Y(s) in Foster expansion is illustrated.

As seen, the proposed IRFA together with the equivalent circuit synthesis method enable generating a highly accurate HF model from a measured impedance (or admittance) with flexibility. Based the IRFA and the circuit synthesis method, the HF model for CM chokes will be introduced in the following section.

IV. IRFA ADAPTED HF MODEL OF CM CHOKES AND EXTRACTION PROCEDURE

A. Proposed HF Model of CM chokes

The proposed HF model of CM chokes is shown in Fig. 7. It should be noted that the parasitic capacitances are assumed to be lossless. Besides, the capacitances C s in Fig. 2(b) are discarded in our model since C e , C p and C c are sufficient for describing the electrostatic behavior of a strongly coupled two-winding transformer [START_REF] Schellmanns | Multiwinding transformers: a successive refinement method to characterize a general equivalent circuit[END_REF]. Instead of using a heuristic method to identify the leakage impedance Z 1 and the magnetizing impedance Z 2 , the topologies of Z 1 and Z 2 of the proposed model is represented by the equivalent circuit in Fig. 6, which are determined by the results of the IRFA method. It is to be noted that the choice of the topologies follows strict mathematical calculation so that the final equivalent circuit is guaranteed to be accurate over a wide frequency range.

B. Extraction Procedure

The extraction of the proposed HF model shown in Fig. 7 begins with three specific impedance measurements T 3 , T 0 and T 2 , as shown in Table II. The IRFA method is then applied to extract the equivalent circuit for the impedances Z T3 , Z T0 and Z T2 . Note that the measurement T 1 in Fig. 3 is discarded since C s is removed from the model. Moreover, the measurement T 4 used in [START_REF] Kotny | High Frequency Model of the coupled inductors used in EMI Filters[END_REF] is not necessary in the proposed procedure because T 2 can provide enough information to extract the magnetizing impedance Z 2 . In order to validate the proposed method and the extraction procedure, it is applied to model a planar CM choke with 8 turns on each winding. First, impedance measurements are carried out with an HP4294A precision impedance analyzer. The configurations T 3 , T 0 and T 2 are treated successively, as described in the following parts. Configuration T 3 represents a capacitive configuration which gives the value of C c +C p . The admittance Y T3 = 1/Z T3 is fitted with the expression where d denotes the dielectric losses and e denotes the capacitance 2(C c +C p ). The fitting starts from 100 Hz to guarantee a good measurement precision of capacitances. As the capacitance model is an R-C parallel cell, the fitting stops at 30 MHz to exclude the resonance, as shown in Fig. 8(a). The loss term d is neglected according to the lossless capacitance assumption. The extracted result for 2(C c +C p ) is 29.98 pF.

0 i r > L R 1 / / i i i L r R p r = = - R C 1 / / i i i C r R r p = = - 0 i r < C R -R ( ) / 1 / i i i R p r C pR = = - L R -R / / i i i R r p L R p = = -

Complex-pole Pair

2 i i i i a s b s ms n + + + Minimal Type L R 1 R 2 C ( ) 2 1 2 1 2 1 1 i i i i i i L a R Lm Lb R n b R C bL R = = - = - = L R 1 C R 2 ( ) ( ) ( ) 1 2 2 1 1 2 1 1 1 i i i i i i C a R Cm C b R n b R L R bC - - = = - = - = Extended Type C 1 L 1 L 2 R 1 R 2 C 2 1 1 1 1 1 2 1 1 2 2 2 2 1 / , 1 1 , 1 , i i i i i L i i R n b C m R L L n C a R mL C n L = = = = - = = L 1 C 1 R 1 L 2 C 2 R 2 1 1 1 1 1 2 1 1 2 2 2 2 / , / 1 1 , 1 1 , i i i i i C i i R b n L R m C C n L a R L m C n C = = = = - = = R1
Configuration T 0 is a flux-subtracting configuration so only the leakage impedance Z l /2 and the capacitance 2(C e +C p ) are concerned. Again, to facilitate the circuit synthesis, the admittance Y T0 = 1/Z T0 is treated in the IRFA. In fact, the capacitance 2(C e +C p ) corresponds the e•s term in Fig. 6, so the rational function Y T0 (s) represented in form of (1) must fulfill: m -n = 1. In order to obtain a simple and accurate equivalent circuit, the values of m and n should be lowest possible but still guarantee an acceptable accuracy. In this case, we choose m = 6 and n = 5 after a few trials. In order to extract the DC resistance as well as the parasitic capacitance 2(C e +C p ), the fitting frequency range covers from 40 Hz to 70 MHz, as illustrated in Fig. 8(b). The fitting results for Y T0 (s) are listed in Table III.

It can be seen from Table III that the extracted e T0 is negative, resulting in a negative capacitance. The latter is not allowed because it will cause simulation instabilities [START_REF] Schellmanns | Equivalent circuits for transformers based on one-dimensional propagation: accounting for multilayer structure of windings and ferrite losses[END_REF]. In fact, the phase plot of Z T0 (see Fig. 8(b)) shows that this impedance is not capacitive beyond the resonance frequency f r ≈ 51 MHz. Around this frequency, the measured impedance Z T0 is affected by distributed effect because the winding length of the studied CM choke is about 1m, which is comparable to the wave length of the FR-4 based PCB structure at f r . Therefore, using the simple parallel circuit shown in Table II to describe Z T0 or Y T0 gives rise to the negative capacitance. To obtain a positive capacitance, an approximation is performed. . ( 14)

The obtained results are: e´T 0 = 3.22E-11 and T0 d ¢ = 1.48E-03, which can be further represented by an R-C parallel cell. With this simplification, the positive parasitic capacitance 2(C e +C p ) and its parallel resistance is obtained. As the complex pole pair is thrown away during the approximation, the fitting precision of the real part of Z T0 is impaired at high frequencies, since the complex poles are indispensable for obtaining high fitting precision around the resonance frequency, as shown in Fig. 9(a). However, trade-offs have to be accepted with this lumped-element equivalent circuit.

The final circuit representation for impedance Z T0 is shown in Fig. 10(b). The impedance Z 1 /2 is contained in the dashed box. The elements of the R-L cells are calculated with the real pole-residue pairs listed in Table III. These components can be interpreted as the Partial Element Equivalent Circuits (PEEC) for modeling the eddy current effect.

Configuration T 2 is used as the last step of the procedure to identify the equivalent circuit of Z T2 . A rough calculation is first performed at 100 kHz to estimate the CM inductance with Im(Z T2 ) = ωL CM /2, giving L CM = 382.4 μH. Similar to

T 0 I 1 I 2 O 1 O 2 Z 1 /2; 2(C e +C p ) 2(C e +C p ) Z 1 /2 T 2 I 1 I 2 O 1 O 2 (Z 1 +Z 2 )/2 ; 2(C e +C c )

2(C e +C c )

Z 1 /2 Z T0 , the admittance form Y T2 is processed for the fitting using a rational function with m = 6 and n = 5. As shown in Fig. 8(c), the fitting stops at 30 MHz, just before the second resonance that cannot be modeled by the lumped-element equivalent circuit in Fig. 7. The fitting results are shown in Table III. According to the equivalent circuit T 2 in Table II, e T2 corresponds to the value of the parasitic capacitance 2(C e +C c ) whereas the remaining parts (i.e. d T2 , real poles and complex poles) represent the admittance 2(Z 1 +Z 2 ) -1 . With Z 1 being identified [see Fig. 10(b)], the magnetizing impedance Z 2 is calculated by subtracting the contribution of Z 1 . The IRFA is applied again to get the pole-residue information of the admittance Y 2 (s)=1/Z 2 (s), as listed in Table III. Note that a real pole p = -2.1616E+08 with negative residue r = -9.4903E+04 is extracted. This negative pole-residue is treated with the method presented in Table I, resulting in a positive R(2.28 kΩ)-C(2.03 pF) series branch, which is interpreted as the nonmagnetic HF losses in the ferrite material due to the capacitive isolation of the grains [START_REF] Schellmanns | Equivalent circuits for transformers based on one-dimensional propagation: accounting for multilayer structure of windings and ferrite losses[END_REF]. The final equivalent circuit of Z 2 /2 and Z T2 is shown in Fig. 10(c

C C C C C C ì ï + = ï ï ï + = í ï ï ï + = ï î . ( 15
)
Till now the HF model for the studied planar CM choke is established. The obtained results will be validated by experimental tests in next section.

V. EXPERIMENTAL VERIFICATION AND DISCUSSION

A. Experimental Verification

To validate the HF model and the procedure, the planar CM choke is first examined by impedance measurements. Two As there are many small-valued elements in the model, the sensitivity of the model should be examined. Some qualitative results are given for this analysis. 1) The three parasitic capacitances C e , C p and C c affect the resonance frequency and the response beyond the resonance. 2) In Z 1 and Z 2 , the branches of dominant poles determine the behavior of the model on the inductive region. The model is less sensitive to the parameters in the braches of non-dominant poles.

3) The impedance around the resonance varies rapidly so they are sensitive to the dominant poles branches, the complex pole-pairs branches and parasitic capacitances.

In a word, the model is quite robust because the behavior of the model does not change abruptly due to small variations of some parameters. However, cares should always be taken during the measurements and the fitting process to guarantee a good accuracy of the extracted model.

Next, an EMI filter using the topology shown in Fig. 1 is realized with the planar CM choke, as shown in Fig. 12(a). The leakage of the CM choke acts as differential mode (DM) inductors. The impedance of the CM capacitors (4.7 pF) and the DM capacitor (68 nF) are measured by impedance analyzer. The equivalent circuit for the capacitors is given in Fig. 12(b) and their impedance Z C (s) can be expressed by: ( )

1 P C E S R E S L P R Z s R sL sCR = + + + (16)
where R ESR stands for equivalent series resistance and L ESL for equivalent series resistance. To get the values of the parameters, ( 16) can be fitted by the IRFA method using a rational function ( 1) with m = 2 and n = 1. The IL of the filter is measured on an Agilent 5071C network analyzer using a four-port S parameter measurement method [START_REF] Kostov | Insertion loss in terms of four-port network parameters[END_REF] [see Fig. where S cc21 and S dd21 give the CM and DM insertion loss for the EMI filter, respectively [START_REF] Kostov | Insertion loss in terms of four-port network parameters[END_REF]. Moreover, a four-port S parameter simulation is performed over the same frequency range with the same configuration as Fig. 12(c). The simulated CM IL S cc21_sim and DM IL S dd21_sim are deduced by [START_REF] Tan | Automatic Identification of Magnetic Component Equivalent Circuits Using Impedance Measurements[END_REF] as well. The comparison between the measured ILs and the simulated ones are presented in Fig. 13. It can be seen that the simulated result closely matches the measurement for common mode. However, large difference at HF is observed for the differential mode. This is due to the parasitic coupling effects among the choke, capacitors and trace loop [START_REF] Wang | Effects of Parasitic Parameters on EMI Filter Performance[END_REF], [START_REF] Wang | Characterization and Parasitic Extraction of EMI Filters Using Scattering Parameters[END_REF], which are not considered in this model. However, this is out of the scope of this paper.

B. Discussion

Through the IRFA method, accurate models for CM chokes can be built from impedance measurements. However, several important aspects of the model and the procedure should be addressed.

1) Passivity:

As the whole extraction procedure treats passive magnetic components, the passivity issues have to be considered. The passivity of a component requires that the component be dissipative in energy. However, the IRFA method cannot guarantee the passivity of the calculated rational function. During the past ten years, many works have been reported on the passivity verification and enforcement techniques [START_REF] Gustavsen | Enforcing passivity for admittance matrices approximated by rational functions[END_REF], [START_REF] Grivet-Talocia | A Comparative Study of Passivity Enforcement Schemes for Linear Lumped Macromodels[END_REF]. In this paper, all the extracted results are verified to be passive using the method of [START_REF] Gustavsen | Enforcing passivity for admittance matrices approximated by rational functions[END_REF]. In practice, as CM chokes are naturally dissipative in HF due to core losses and copper losses, the passivity is usually respected if fitting precision is good enough.

2) Distributed effect: Due to the distributed effect, manual adjustments are needed to avoid negative elements, which require some experience about the equivalent circuit synthesis. This is the limit of using a lumped-element equivalent circuit model. The model presented in this paper is effective up to 30 MHz, which corresponds to the frequency range of interest for the conducted emissions. Nonetheless it is necessary to consider the distributed effect to cover frequencies beyond 30 MHz, which requires implementing a distributed circuit model.

3) Saturation: CM chokes always work under flux bias which will cause saturation effect [START_REF] Zhu | An Integrated AC Choke Design for Common-Mode Current Suppression in Neutral-Connected Power Converter Systems[END_REF]. The proposed model is established on the basis of small signal measurements. Therefore, the error of the model will increase as the current increases. However, the primary focus of the paper is to propose a small signal HF model of CM chokes that can be systematically synthesized. As the first step, the small signal model is now available and it can then be coupled with nonlinear cell that accounts for the saturation of the magnetic material, which is the objective of a future study.

VI. CONCLUSION

In this paper, a HF equivalent circuit model of CM chokes used in EMI filter is proposed incorporating a systematic extraction procedure. The procedure uses an iterative rational function approximation method to extract pole-residue form expressions from impedance measurements. Based on the pole-residue information, a systematic synthesis method is applied to generate the equivalent circuit. The proposed procedure gives accurate models in a short time. This procedure is applied on a planar CM choke and is validated by simulations and measurements.

APPENDIX

According to the SK iteration, the denominator of the iteration t-1 D [t-1] (s k ) is divided at both sides of the equation (3) of the iteration t, resulting in [START_REF] Wu | Design, Modeling, and Improvement of Integrated EMI Filter With Flexible Multilayer Foils[END_REF]

where the superscript * denotes the complex conjugate. For magnetic components, the imaginary part of the impedance (Z≈ωL) is much larger than its real part (losses) in the inductive region. This unbalance will lead to a low fitting quality for the real part during the least square process. In order to achieve a balanced fitting precision between the real part and the imaginary part, the real and imaginary parts of ( 18) are normalized respectively by 1/Re[Z [t-1] (s k )] and 1/Im[Z [t-1] (s k )] as weighting factors:

(

Combining the SK iteration [START_REF] Elzinga | Pole-residue formulation for transient simulation of high-frequency interconnects using householder LS curve-fitting techniques[END_REF] with the weighting factors [START_REF] Levy | Complex-curve fitting[END_REF], the final formulations [START_REF] Tarateeraseth | Accurate Extraction of Noise Source Impedance of an SMPS Under Operating Conditions[END_REF] and ( 8) are derived.

AFig. 1 .

 1 Fig. 1. Typical topology of EMI filter

Fig. 2 .

 2 HF circuit model for CM chokes proposed in [15]. (a) Toroidal CM choke. (b) HF equivalent circuit. (c) Equivalent circuit of leakage impedance Z1. (d) Equivalent circuit of magnetizing impedance Z2.

Fig. 3 .

 3 Fig. 3. Impedance measurement configurations for the extraction. (a) T0. (b) T1. (c) T2. (d) T3. (e) T4.

3 . 6 .Fig. 4 .

 364 Fig. 4. IRFA algorithm.

Fig. 5 (

 5 Fig. 5(b) and (c), the IRFA methods alleviate the frequency bias problem in Fig. 5(a). As seen, better fitting accuracies are achieved at LF. The difference between Fig. 5(c) and (b) is the implementation of balanced weighting. It can be seen from the error curve shown in Fig. 5(c) that with balanced weighting, the fitting accuracy of the real part is further improved in comparison with Fig. 5(b).The precision of IRFA fitting is very good on wide frequency band because the impedance curve is quite smooth (very few resonances). However, the error may exceed the specified error level ε max on the two extremities of fitting range and the resonance frequencies. To achieve a better precision, one can increase the order of the rational function at the expense of obtaining a more bulky equivalent circuit. In order to obtain a simple equivalent circuit, the choice of the order turns out to be a trade-off between the accuracy and the complexity of the circuit.

Fig. 5 .

 5 Fig. 5. Comparison of the fitting results. (a) RFA with frequency bias problem. (b) IRFA without balanced weighting. (c) IRFA with balanced weighting.

Fig. 7 .

 7 Fig. 7. Proposed HF model for CM chokes.

Fig. 6 .

 6 Fig. 6. Equivalent circuit synthesis for the Foster expansion of Y(s).

  First, the complex pole-residue pair ( T0 p , T0 r )-( * T0 p , * T0 r ), d T0 = 3.0893E-3 and e T0 = -3.2803E-12 terms of Y T0 (s) in Table III are recombined together to form a new admittance called Y CT0 (s) "true" parasitic capacitance is merged in Y CT0 (s). The expression (13) is then simplified using e¢ s by equating the values of Y CT0 (s) and frequency s r = j2πf r , as given by

Fig. 8 .

 8 Fig. 8. Measured impedances for the extraction. (a) ZT3. (b) ZT0. (c) ZT2.

  ). In Z 2 , the L(190 μH)-R(4 mΩ) branch presents the dominant pole according the definition: a pole p i with its residue r i is called dominant pole if its F i = |r i /Re(p i )| is much larger than the other poles. The values F i for the poles of Z T0 and Z 2 are listed in TableIIIand the F i of the L(190 μH)-R(4 mΩ) branch is much larger than the others. Therefore, it has the most important influence in Z 2 . The 190 μH inductance is in fact the CM filtering inductance because it is half the CM inductance L CM estimated from Z T2 . The simulated impedance of Z T2 is compared with measured data in Fig.9(b) in terms of real part and magnitude. As seen, the simulated results agree well with the measurements up to 30 MHz.With the parasitic capacitances obtained from the three configurations, three linear equations expressed in (15) can be established, giving the values of the stray capacitances: C c = 6.1 pF, C e = 7.2 pF and C p = 8.9 pF.

Fig. 9 .Fig. 10 .

 910 Fig. 10. Extracted circuits for the CM choke. (a) Z T3 . (b) Z T0 . (c) Z T2 .

12

 12

  (c) and (d)]. The S parameter matrix [S ij , i,j=1,2,3,4] is measured from 100 kHz to 30 MHz with logarithmic sweep. Postprocessing of this matrix is performed to obtain the mixedmode S parameter S cc21 and S dd21 through the following equations:

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Comparison between the measured and simulated impedances of the HF model. (a) Open circuit test-ZO. (b) Short circuit test-ZSC.

Fig. 13 .

 13 Fig. 13. Comparison of simulated and measured IL. (a) CM. (b) DM.

TABLE I EQUIVALENT

 I CIRCUIT SYNTHESIS FOR REAL POLES AND COMPLEX-POLE PAIRS

	Term	Circuit	Y(s)	Expression	Circuit	Z(s)	Expression
	Real pole
	i r
	i s p -

TABLE II MEASUREMENT

 II CONFIGURATIONS FOR THE PROPOSED EXTRACTION PROCEDURE

	Notation	Measurement Configuration	Element	Equivalent Circuit
	T 3	I 1 I 2 O 1 O 2	2C p +2C c	2(C p +C c )

TABLE III FITTING

 III RESULTS FOR THE PLANAR CM CHOKE

Y T0 (s) = 1/Z T0 (s) Y T2 (s) = 1/Z T2 (s) 2Y 2 (s) = 2/Z 2 (s) d

  T0 = 3.0893E-3; e T0 = -3.2803E-12 d T2 = 5.2701E-04; e T2 = 2.6690E-11 d 2 = 4.9593E-04; e 2 = 8.2615E-14

	Poles	Residues	Fi	Poles	Residues	Poles	Residues	Fi
	-7.3080E+07	3.6178E+06		-1.6659E+07	-2.1578E+02	-1.6679E+07	-2.1486E+02
	+i5.5497E+08 -7.3080E+07	+i1.2356E+06 3.6178E+06	0.05	+i1.8259E+07 -1.6659E+07	+i6.3580E+02 -2.1578E+02	+i1.8265E+07 -1.6679E+07	+i6.3909E+02 -2.1486E+02	4.0E-5
	-i5.5497E+08	-i1.2356E+06		-i1.8259E+07	-i6.3580E+02	-i1.8265E+07	-i6.3909E+02
	-2.8458E+07	1.0814E+05	3.8E-3	-2.2332E+08	-1.0501E+05	-2.1616E+08	-9.4903E+04	4.4E-4
	-1.8105E+06	1.6114E+05	0.09	-2.0256E+05	4.2972E+01	-2.0165E+05	4.2780E+01	2.1E-5
	-2.3224E+05	3.0680E+06	13.2	-4.1835E+02	5.2350E+03	-2.4875E+01	5.2440E+03	211


Dr Jean-Luc Kotny
email: jean-luc.kotny@univ-lille1.fr

Dr Xavier Margueron
email: xavier.margueron@ec-lille.fr

Member, IEEE Nadir Idir
email: nadir.idir@univ-lille1.fr

This paper deals with a high frequency modeling method of the coupled inductors used in EMI filters. These filters are intended to reduce conducted emissions generated by power static converters towards the power grid.

To model the EMI filters, it is necessary to identify the various parameters of the passive elements: inductors and capacitors. Because of their major impact on filter efficiency, these elements must be identified with accuracy. In this study, high frequency model of common mode coupled inductors is proposed. The identification of the model parameters is based on the experimental approach. Simulation results of the proposed model are compared to the experimental data obtained using the specific experimental setup. These results made it possible to validate the EMI filter model and its robustness in a frequency range varying from 9 kHz to 30 MHz. The proposed high frequency inductor models will be very helpful for design and optimization of EMI filters, since the high frequency behavior of the filter mainly depends on magnetic materials used and on the geometrical characteristics of winding.

High Frequency Model of the coupled inductors used in EMI Filters

I. INTRODUCTION

Nowadays, the increase of power converter number is very problematic for electromagnetic compatibility (EMC). Indeed, each power static converter generates a lot of high frequency (HF) interferences toward the power network which becomes more and more polluted [START_REF] Wang | Investigation of Hybrid EMI Filters for Common-Mode EMI Suppression in a Motor Drive System[END_REF]. The main solution to reduce these conducted emissions is based on the utilization of EMI filters [START_REF] Akagi | Design and performance of passive EMI Filter For Use with A Voltage-Source PWM Inverter Having Sinusoidal Output Voltage And Zero Common-Mode Voltage[END_REF]- [START_REF] Xiaofeng | Integrated EMI Filter Design with Flexible PCB Structure[END_REF]. It can also be combined with other solutions like the slowing down the dv/dt and di/dt during the transitions of the power semiconductor components and/or by acting on the converter controls [START_REF] Idir | Active gate voltage control of turn-on di/dt and turn-off dv/dt in insulated gate transistors[END_REF]- [START_REF] Videt | A New Carrier-Based PWM Providing Common-Mode-Current Reduction and DC-Bus Balancing for Three-Level Inverters[END_REF].

The EMI filters are made of coupled inductors combined with capacitors; the choice of the topology depends on network and load impedances. A typical differential mode (DM) and common mode (CM) filter used for power converter is shown in Fig. 1. It is well-known that magnetic components have a strong impact on filter efficiency [START_REF] Chen | Modeling of parasitic inductive couplings in a Pi-shaped common mode EMI filters[END_REF]- [START_REF] Chen | Improving the Characteristics of Integrated EMI Filters by Embedded Conductive Layers[END_REF]. They are typically made with copper wiring on high permeability ring ferrite. These chokes are traditionally designed by a "cut and try" process, which is very expensive and time consuming because of the high number of prototypes which are needed. Wires specifications and their placement, PCB characteristics, or magnetic core performances introduce parasite effects that can reduce drastically the filter's efficiency [START_REF] Wu | Design, Modeling, and Improvement of Integrated EMI Filter With Flexible Multilayer Foils[END_REF]- [START_REF] Leferink | High Performance EMI Filter for Frequency Converters[END_REF].

The stray elements of the passive components, such as equivalent series inductor (ESL) of capacitors and equivalent parallel capacitor (EPC) of inductors, degrade the performances of the EMI filters. While for a lot of cases, their influence is less important than the couplings among components and layouts [START_REF] Shuo Wang | Effects of Parasitic Parameters on the Performance of EMI Filters[END_REF].

In order to design and optimize the performance of the filter by simulation, a high frequency model, including parasitic elements, must be used. These elements must be identified with a good accuracy [START_REF] Wang | Characterization and Parasitic Extraction of EMI Filters Using Scattering Parameters[END_REF]. In this paper, a high frequency modelling method of the coupled inductors of EMI filter is presented. The new equivalent circuit is detailed and its characterisation process is described. The obtained model is tested with different magnetic material and gives good results on a large frequency range from 1 kHz to 30 MHz. In order to validate the proposed model and to test its reliability, a common mode inductors and capacitors are used to realise an EMI filters. The simulation results using the high frequency model of the filter are validated by experimental measurements. In the following section, high frequency modelling method of the coupled inductors is detailed.

II. HIGH FREQUENCY MODEL OF COUPLED INDUCTORS

In this study, a prototype of the coupling inductors has been realized with EPCOS ferrite material N30 [12]. The core shape is a ring R58.3/40.8/17.6 and each winding is made of 15 turns of copper (diameter 1.5mm). At first, we will propose an equivalent circuit model of the coupled inductors. Then, we propose an experimental method of characterization of the passive component which will enable to identify the model parameters.

A. Equivalent Circuit of the coupled inductors

Coupled inductors used in EMI filters are a specific case of traditional n-windings magnetic components [START_REF] Margueron | Design of Equivalent Circuits and Characterization Strategy for n-Input Coupled Inductors[END_REF]- [START_REF] Margueron | Identifying the Magnetic Part of the Equivalent Circuit of n-Winding Transformer[END_REF] because they are made with two identical windings around a magnetic core (Fig. 2a). The objective is to have a model of coupled inductors based on equivalent electrical circuit that enable to describe the various physical phenomena (magnetic and electrostatic) which appear in high frequency. In order to have a more precise model, a symmetrical equivalent circuit of these inductors, is used (Fig. 2b). The proposed model is a lumped model; it uses only R, L, and C elements to describe its behaviour in a large frequency range. Half of the equivalent circuit is made of three impedances It is known that for a 2-windings transformer model, with a strong magnetic coupling, only three capacitors are necessary to describe its electrostatic behaviour [START_REF] Margueron | Identifying the Magnetic Part of the Equivalent Circuit of n-Winding Transformer[END_REF]. In the case of the coupled inductors and in order to have a symmetric model (for each winding), a supplementary capacitor C S has been added to the circuit as shown in Fig. 2b.

B. Parameters identification method

In order to determine the parameters of the common mode inductors model (Fig 2b), impedance measurements realized in different test configurations are necessary. An impedance analyzer (Agilent 4294A) is then used and it enables to obtain impedance measurements (modulus and phase) on a large frequency range from 1 kHz to 100 MHz [START_REF]Agilent 4294A Precision Impedance Analyzer-Operation Manual[END_REF].

To identify the various impedances of magnetic component, different test configurations can be carried out [START_REF] Wang | Improvement of EMI filter performance with parasitic coupling cancellation[END_REF]. Due to the geometrical symmetry of coupled inductors and using the equivalent circuit presented in the Fig. 2b, only five well-chosen measurements, as shown in Fig. 3, should be sufficient for determine all the magnetic and electrostatic parts of the model.

In order to determine all the parameters of the coupled inductors model, different kind of impedance are necessary. For example, subtracting flux or additional flux measurements are realized and only one open circuit measurement is used for the magnetizing inductance identification. Figure 3 describes the main five measurement configurations (T 0 until T 4 ) that are chosen to identify the equivalent circuit elements of the coupled inductors.

In the determination process of the model parameters, the first configuration T 0 is used to identify the leakage impedance 1 Z .

Then, T 0 , T 1 , T 2 and T 3 are combined to calculate the four capacitances and finally T 4 enables to identify the magnetizing part of the component. The Table I sum up the various measurement configurations which allow to identify the model parameters. The five impedances measured on the prototype using the impedance analyzer are shown in Figures 4 and5. The impedance measured in configuration T 3 is quite different from the others because it corresponds to a capacitive behaviour from 1 KHz to 30 MHz. Indeed, this impedance is measured between both windings short circuited so the measurement is equivalent to a capacitance.

III. DETERMINATION OF THE MODEL PARAMETERS

A. Leakage Impedance

The leakage impedance parameters are obtained from the measurement configuration T 0 because windings are connected in a subtracting flux topology (cancellation of magnetizing flux). The impedance obtained from T 0 corresponds to parallel RLC impedance which the equivalent circuit is shown in Fig. 6. The inductance 1 L is identified in frequency band (between 1 kHz and 3.7 MHz) when the impedance measured in this configuration increases (and phase equal to 90°). The inductance variation (with frequency) is due to eddy current effects in the copper wires. Until the resonance frequency, the error between the model and the measurement is still lower than 5%. The obtained model is quite simple but can be improved using ladder RL circuits in order to model correctly the eddy current effects [START_REF] Margueron | Identifying the Magnetic Part of the Equivalent Circuit of n-Winding Transformer[END_REF]. The value of the resistance R 1 is measured at the resonance frequency

( MHz f r 9 . 19 
) of the impedance obtained from T 0 as shown in Fig. 4. The capacitance of the equivalent circuit of the coupled inductor in configuration T 0 is given by

) ( 2 0 Cp C C e   .

B. Parasitic Capacitances

The capacitance value C 0 is identified from the leakage impedance measured in configuration T 0 . However, it remains three other capacities to calculate. With three more equations, the system can be solved. These equations are given by measurements carried out in configurations T 1 , T 2 and T 3 . Their equivalent circuits are presented in Fig. 7. The impedance measured in configuration T 1 is quite similar to T 0 . Thus, the model used is also a parallel RLC circuit (Fig. 7a). The measured values of R and L are four times higher than the leakage parameters. The capacitance value C 1 is also determined from the resonance frequency. The value of this capacitance is given by the following relation:

pF C C C C C e C p S 8 . 6 2 1      (2)
The equivalent circuit of the impedance measured in configuration T 2 cannot use only one RLC series circuit. Indeed, one will show, in the next section, that the magnetizing impedance will require three RLC circuits to describe its high frequency behaviour. Because the focus is on the electrostatic part of the impedance, for frequency upper than 50MHz, a tangent of the impedance curve is plotted. The obtained equivalent capacitance is equal to:

pF C C C C e 8 ) ( 2 2    (3)
The last needed equation is given by impedance measured in the configuration T 3 . Its equivalent representation is a RLC series circuit shown in the Fig. 7c. The capacitance is easily evaluated between 1 kHz and 4 MHz using the imaginary part of the measured impedance. The equivalent capacitance is equal to:

pF C C C C S C P 2 . 19 ) ( 2 3     (4)
Finally, these measurements lead to a 4-unknown equations system (5) that can be easily solved and the obtained results are given in [START_REF] Chen | Modeling of parasitic inductive couplings in a Pi-shaped common mode EMI filters[END_REF].

                  3 2 1 0 2 2 2 2 2 2 2 2 2 C Cs Cp Cc C Cc Ce C Cs Cp Ce Cc C Cp Ce (5)                           2 2 2 2 4 2 2 4 2 4 3 1 0 3 2 1 0 3 2 1 3 2 1 C C C Cs C C C C Cp C C C Ce C C C Cc (6)
The table III sums up the obtained values for the four capacitances.

TABLE III CAPACITOR VALUES

One can note that the value of the capacity C e is negative. This can be justified by the simplicity of the proposed behaviour model of the impedances and also to the measurement errors related to the accuracy of the impedance analyzer (measurement equipment).

C. Magnetizing Impedance

The impedance measured in configuration T 4 enables to characterize the magnetizing part Z 2 of the component. Thus, such impedance, shown in Fig. 8, cannot be described by only one RLC circuit. Three different circuits have to be used in order to describe the three areas. -The first area, from low frequency to F 1 = 670 kHz, corresponds to the proper inductance traditionally calculated using the following relation:

L A N L   2 (7)
With:

N: winding turns, A L : inductance factor.

The inductance factor of the ferrite material N30, used to realized a prototype of the coupling inductors, is equal to A L =5400 ± 20%. Thus, it's well known that the characteristics of magnetic materials used in the design of the EMI filters depend on the levels and frequencies ranges of the conducted disturbances to be attenuated.

-The second area, from F 1 to F 2 = 11 MHz, corresponds to high frequency effects of the magnetic material. This frequency band is divided into two parts 2a and 2b. At the frequency F' 2 , one can note a slope break of the impedance modulus, thus, the magnetic material effect is masked by an electrostatic effect.

-In the third area, a resonance appears which can be linked to the inductive leakage phenomena combined with electrostatic effects.

The magnetizing impedance evolution measured in configuration T 4 is shown in Fig. 8. The equivalent circuit at this impedance, which corresponds only to the area 1 and 2, is shown Fig. 9a. In this frequency band (area 1 and 2), the influence of the leakage inductances and the parasitic capacitances is negligible. The value of L 2 is calculated using the imaginary part of the impedance (L 2 = Z 2 sin ) in the frequency band between 1

KHz and 370 kHz as shown in the Fig. 8 (or using relation ( 7)) . The obtained value of this inductance is equal to L 2 = 1.32mH.

The elements R' 2 and C 2 are obtained from the first resonance frequency F 1 =670 KHz (Fig. 8). The resistance value is equal to R' 2 = 7340. The capacitance 2 C is calculated with the following relation:

  pF F L C 4 . 21 2 2 1 2 1 2 2       (8)
The elements R 2 and R C2 fix the impedance slope between the frequencies F 1 and F' 2 (area 2a). The value of the resistance R C2 = 4980 is obtained from the impedance slope in this area. However, the resistance R 2 is calculated using the equivalent resistance at the frequency F' 2 =3 MHz. Its value is equal to R 2 = 2026.

The last equivalent circuit L' 2 and C' 2 is used to model the evolution of the impedance between F' 2 and F 2 (area 2b). The values of these parameters are obtained by a fitting method using APLAC software. In order to show the influence of the various elements on the accuracy of the proposed model, we present in Fig. 10a the comparison between a simulation of the equivalent circuit of the magnetizing impedance without elements R2 and RC2. However, Fig. 10b show the simulation of the same circuit without elements L'2 and C'2. IV. The obtained model uses only electrical equivalent circuits that enable to make simulations in frequency and time domains using simulator software such as SPICE. However, for time domain simulation, negative capacitances must be removed to positive one using inverse perfect couplers [START_REF] Margueron | Identifying the Magnetic Part of the Equivalent Circuit of n-Winding Transformer[END_REF]. The coupled inductors are associated with capacitors to achieve a complete EMI filter. Components are connected together on the PCB and placed inside a metallic box as shown in Fig. 14. Figure 15 shows the interior of the box which contains the common mode inductors and four capacitors C Y = 4,7nF. The two small boxes are the resistive dispatchers which used to divide the power injected in an equal way on the two inputs ports of the filter. For simulation, the capacitors "C Y " have been characterized using the impedance analyzer. The equivalent circuit used is a RLC series circuit:

nF C 9 . 4  ,   18 . 0 R , . 4 . 6 nH R 
The connection wires were also modeled [START_REF] Grandi | Effects of power converter parasitic components on conducted EMI[END_REF]- [START_REF] Zhu | Characterization of power electronics system interconnect parasitics using time domain reflectometry[END_REF].

Both common mode and differential mode filters have been tested according to experimental setup shown in Fig. 16. A spectrum analyzer (HP ESA L1500A) is used to characterize the common mode and the differential mode filters. In the differential mode configuration test, a high frequency balun (transformer) is added in order to obtain a balanced system. The

Tracking RF generator power level (output side) is equal to 3dBm for the common mode characterization and 0dBm for the differential mode characterization. The filters are characterised with input/output impedances equal to 50/50.

The both configuration filter attenuations are calculated by simulation using the different components models (coupled inductors, capacitor C Y , wires…). Figures 17 and18 show comparison between measured and calculated attenuation for common mode and differential mode filters. For both configurations, these results show a low difference (3 dB) between measurement and simulation up to 30 MHz.

The filter attenuation depends mainly on the magnetic material and capacitors characteristics [START_REF] Neugebauer | Parasitic capacitance cancellation in filter inductors[END_REF]- [START_REF] Caponet | EMI filters design for power electronics[END_REF]. On the other hand, the maximum attenuation strongly depends on the filter realization (L and C positioning, coaxial wire lengths, connections lengths…). In conclusion, filter efficiency mainly depends on its component characteristics and their relative positioning, so a particular care must be taken while realizing an EMI filter for reducing these parasitic effects.

Fig. 1 .

 1 Fig. 1. Typical EMI Filter topology

  Fig. 2. High frequency model of the coupled inductors

Fig. 3 .

 3 Fig. 3. Impedance measurement configurations

Fig. 4 .Fig. 5 .

 45 Fig. 4. Impedance (modulus) measurement configurations (T 0 , T 1 , T 4 )

  All impedances are modeled by one or more combination of RLC circuit. The aim of this section is to propose equivalent circuits which allow to model the frequency variation of leakage impedance ( 1 Z ), magnetizing impedance ( 2 Z ) and the 4 capacitors ( c C , e C , p C and s C ) of the coupled inductors shown in Fig. 2b.

Fig. 6 .

 6 Fig. 6. Equivalent circuit of the measurement configuration (T 0 ) From the curve (T 0 ) shown in Fig. 4, the capacitance 0 C is computed from the resonance frequency ( MHz f r 9 . 19  ) using equation (1). This capacitance corresponds to the combination of both capacitances e C and p C . The table II sums up these leakage parameter values.

Fig. 7 .

 7 Fig. 7. Equivalent circuit of three measurement configurations

Fig. 8 .

 8 Fig. 8. Impedance modulus in open circuit (configuration (T4))

Fig. 9 .

 9 Fig. 9. Modelling of the magnetizing impedance

2 Fig. 10 .

 210 Fig. 10. Model of the magnetizing impedance The comparison between simulation results and measurement data (Fig 9b) shows that the proposed equivalent circuit allows to model, with a good accuracy, the evolution of the magnetizing impedance Z 2 . Figure 11 presents the final high frequency coupled inductors model. Its parameter values are listed in the Table IV. The obtained model uses only electrical equivalent

Fig. 11 .

 11 Fig. 11. High frequency model of the coupled inductors

  Fig. 12. Impedance measurement configurations

Fig. 14 .Fig. 15 .

 1415 Fig. 14. EMI filter characterization

Fig. 17 .Fig. 18 .

 1718 Fig. 17. Common mode filter response. Tracking generator power level (output) is equal to 3dBm

TABLE I MODEL

 I PARAMETERS IDENTIFICATION

TABLE II LEAKAGE PARAMETERS

 IIPARAMETERS