A High Frequency Equivalent Circuit and Parameter Extraction Procedure for Common Mode Choke in the EMI Filter

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 3, MARCH 2013 1157

A High Frequency Equivalent Circuit and ParameterExtraction Procedure for Common Mode

Choke in the EMI FilterWenhua Tan, Carlos Cuellar, Member, IEEE, Xavier Margueron, Member, IEEE, and Nadir Idir, Member, IEEE

Abstract—Power converters with high switching frequency gen-erate conducted electromagnetic interference (EMI) noise. EMIfilters are thus widely used to reduce these conducted noises for thecompliance with electromagnetic compatibility standards. In thispaper, a high-frequency (HF) equivalent circuit model for commonmode (CM) chokes used in EMI filters is proposed together withits parameter extraction procedure. This procedure is based onimpedance measurements and it incorporates an iterative rationalfunction approximation fitting algorithm to extract the parame-ters in the model. The proposed model and procedure is appliedto a planar CM choke which is used to realize an EMI filter. Thesimulated results of the filter show good agreement with the ex-perimental ones. This extraction procedure is quite general andit can also be extended to identify the HF model of other passivecomponents.

Index Terms—Common mode (CM) choke, electromagneticinterference (EMI) filter, equivalent circuit, rational functionapproximation.

I. INTRODUCTION

E LECTROMAGNETIC interference (EMI) filters are com-monly used solutions for mitigating the conducted emis-

sions produced by power converters [1]. A typical topology ofEMI filter with ideal components is shown in Fig. 1. However, areal EMI filter is much more complicated and its performancesdepend on many factors such as filter topology, magnetic ma-terial, grounding method, stray elements of components, andparasitic coupling between components etc. [2]–[4]. Completephysic-based models of an EMI filter or their components re-quire extensive electromagnetic analysis, so they are very com-plex and specific to limited applications [5], [6]. Modeling meth-ods based on measurements are more general, direct, and accu-rate for describing the characteristics of the device under study.

The insertion loss (IL) of an EMI filter is usually measuredwith 50 Ω/50 Ω convention. However, the IL of the filter underworking condition depends on the impedances of the source and

Manuscript received February 6, 2012; revised March 30, 2012 and May25, 2012; accepted July 2, 2012. Date of current version October 12, 2012.Recommended for publication by Associate Editor K. Ngo.

W. Tan and X. Margueron are with the Laboratory of Electrical Engineeringand Power Electronics of Lille, Ecole Centrale de Lille, Villeneuve d’Ascq59655, France (e-mail: [email protected]; [email protected]).

C. Cuellar is with the Universite des Sciences et Technologies de Lille,Villeneuve d’Ascq 59655, France (e-mail: [email protected]).

N. Idir is with Instituts Universitaires de Technologie A, University of Lille1, Villeneuve d’Ascq 59655, France (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPEL.2012.2209206

Fig. 1. Typical topology of the EMI filter.

the load, which vary with frequency [7], [8]. In order to obtainthe IL of the filter, many solutions have been reported. In [9], afour-port measurement method using a vector network analyzeris presented. This method consists in a black-box modeling ofthe whole filter with mixed-mode S-parameters. By postprocess-ing the obtained S-parameter data, the IL of the filter can hencebe calculated with any source and load impedances. In [10], amodal model of common mode (CM) chokes based on four-portS-parameter measurements is presented. Though modal modelsenable to analyze the conversion of the noise between differen-tial mode (DM) and CM, they also require compatible modalmodel of noise sources for simulations, which complicates themodeling process. An alternative is to use equivalent circuitmodels, which are physic based and are compatible with mostof the simulation tools. In [11], the equivalent circuit of a DMEMI filter is identified by S-parameters measurements. This ap-proach can correctly extract the parasitic couplings in the filter,resulting in good modeling precision. Impedance measurementis a more frequently used technique, which has long been studiedfor identifying the equivalent circuits of passive magnetic com-ponents [12], [13]. Recently, a lumped-element high-frequency(HF) model for CM chokes has been proposed [14], [15]. Thismodel can be easily built by extracting the parameters fromimpedance measurement results and can effectively describethe HF characteristics of the studied the CM choke and EMIfilter. However, the topology of the HF model is chosen heuris-tically, which requires lots of experience and tests. Moreover, theextraction procedure derives the parameters of the model by ob-serving the impedance curves, leading to quite time-consumingtrial/error iterations.

To simplify the extraction process, numerical methods can beapplied. In [16], a genetic algorithm is used to extract the behav-ior model of chokes, yielding an accurate and reliable broadbandequivalent circuit. Nevertheless, the convergence time of suchalgorithm is usually long when the problem has a large searchspace. In [17], a physics-based equivalent circuit of CM chokes

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1158 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 3, MARCH 2013

Fig. 2. HF circuit model for CM chokes proposed in [15]. (a) Toroidal CMchoke. (b) HF equivalent circuit. (c) Equivalent circuit of leakage impedanceZ1 . (d) Equivalent circuit of magnetizing impedance Z2 .

is built with a rational function approximation (RFA) methodbased on linear optimization [18]. However, the topology of theequivalent circuit is predefined so prior knowledge about thecomponent is needed.

In this study, an HF model of CM chokes is proposed togetherwith a systematic extraction procedure. This procedure is aidedby the RFA method incorporating an iterative scheme (referredas IRFA) and it allows a fast and accurate extraction of theparameters in the model. This paper is organized as follows.In Section II, the HF model introduced in [15] is reviewed.Next, the RFA and the proposed IRFA methods are presented inSection III. In Section IV, the proposed HF model of CM chokesand its extraction procedure are introduced and then applied toa planar CM choke. Finally, experimental verifications of theprocedure as well as some discussion of the results are given inSection 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[15] [see Fig. 2(a)] is illustrated in Fig. 2(b). As seen, it is asymmetrical structure and the transformation ratio η is assumedto be unity. This model can describe the behavior of many typesof CM chokes as long as the winding symmetry is satisfied. Theleakage impedance Z1 includes the winding resistance, the leak-age inductance, and their variations due to skin and proximityeffects of the winding conductor [see Fig. 2(c)]. The magne-tizing impedance Z2 describes the behavior of magnetic core,including the frequency-dependent inductance and losses [seeFig. 2(d)]. In addition, Ce stands for the self-parasitic capaci-tance of each winding whereas the sum Cs + Cp + Cc is relatedto the interwinding capacitances of the component. Examiningthe admittance Y2 = (Z2)−1 , it is found that Y2 contains:

1) one pole at origin: the branch of L2 ;2) one stable real pole: the branch of R2 , RC 2 , C2 , and R′

2 ;3) one pure imaginary pole-pair: the branch of L′

2 and C ′2 .

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

With these elements, the HF behavior of the magnetizingimpedance Z2 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 pre-sented in Fig. 3. All the parameters of the model are obtainedthrough these impedance measurements summarized as follows:

1) T0 , T1 , T2 , and T3 : parasitic capacitances Cc , Ce , Cp , andCs ;

2) T0 and T1 : leakage impedance Z1 ;3) T2 and T4 : magnetizing impedance Z2 .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. Thevalues of the parasitic capacitances are calculated using theresonance frequencies. To fit the model with the measurementresults, some manual adjustments of the parameters are neededfor Z1 and Z2 .

Though good accuracy is achieved in [15], the HF modelshown in Fig. 2(b) can be improved for the following reasons:

1) The extraction procedure requires manual adjustments onthe 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 ac-cording to experimental observations. For example, theadmittance Y2 = 1/Z2 is chosen to have one stable poleand one imaginary pole-pair from experience. In conse-quence, a systematic equivalent circuit synthesis methodis needed to choose the topology.

In view of these necessities, an HF equivalent circuit modelfor CM chokes and an IRFA-based extraction procedure areproposed to facilitate the process. The IRFA method is firstpresented 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 circuitnetwork can be represented by a rational function

Z(s) =N(s)D(s)

=bm sm + bm−1s

m−1 + · · · + b1s + b0

ansn + an−1sn−1 + · · · + a1s + 1(1)

TAN et al.: HIGH FREQUENCY EQUIVALENT CIRCUIT AND PARAMETER EXTRACTION PROCEDURE 1159

with |m − n| ≤ 1 and s = jω. Without loss of generality, onlythe impedance Z(s) is considered in this section. The sameanalysis is valid for the admittance Y (s). The goal of the RFAis to search the values of ai and bi to minimize the error be-tween the rational function Z(s) and measured data Zmeas(s),as expressed by

arg minai ,bi

∑

k

|Zmeas(sk ) − Z(sk )|2 ⇒ Zmeas(sk ) =N(sk )D(sk )

(2)where sk = jωk . To handle this nonlinear optimization problem,Levy used a linearization technique [19]. Stemming from Levy’smethod, the authors introduced in [18] the RFA method forgenerating the macromodel of HF interconnects. Reformulating(2) by multiplying D(sk ) gives

N(sk ) − Zmeas(sk )D(sk ) = 0. (3)

Separating the real parts and imaginary parts of (3) yields alinear system of ai and bi

{Re[N(sk )] − Re[Zmeas(sk )D(sk )] = 0

Im[N(sk )] − Im[Zmeas(sk )D(sk )] = 0.(4)

This system is overdetermined due to the large number of mea-sured points. In order to calculate the unknowns ai and bi , aleast-square method using QR factorization is applied to solvethe overdetermined system (4). Once the values of bi are known,the poles of (1) can be derived by solving the roots of the de-nominator D(s). The impedance Z(s) can then be written intoa pole-residue form

Z(s) = d + e · s +n∑

i=1

ri

s − pi. (5)

It should be noted that ri and pi can be complex-valued. Again,the terms d, e, and ri are solved from this linear overdeterminedsystem (5) by the least-square method [18].

B. Iterative Rational Function Approximation Method

The RFA method is proven to be efficient with noise-freesimulated data. However, it cannot be directly employed for ourapplication, since Levy’s linearization technique is known tohave a frequency bias problem [20]. In (2), Zmeas(sk ) is mul-tiplied by D(sk ) that increases rapidly with sk (as frequencyincreases). This means the errors of (2) are biased by D(sk ),making HF errors more important than low frequency (LF) er-rors. As a result, this frequency bias due to D(sk ) causes poorfitting precision at LF. However, to correctly identify elementslike dc winding resistance of CM chokes, this lack of fittingprecision at LF is undesirable. To handle the frequency bias, theRFA method is improved in this study using the Sanathanan–Koerner (SK) iteration [20]. The denominator of the iterationt − 1D[t−1](sk ) is divided on both sides of (3) for the iterationt, giving

N [t](sk )D[t−1](sk )

− Zmeas(sk )D[t](sk )D[t−1](sk )

= 0. (6)

Fig. 4. IRFA algorithm.

It can be seen that the bias due to D[t](sk ) is alleviated bydividing D[t−1](sk ), resulting in better fitting accuracy at LF.Based on (4) and (6), the proposed IRFA formulation used inthis study is given by{

Re[P [t]Re(sk )N [t](sk )] − Re[P [t]

Re(sk )Zmeas(sk )D[t](sk )] = 0

Im[P [t]Im (sk )N [t](sk )] − Im[P [t]

Im (sk )Zmeas(sk )D[t](sk )] = 0

(7)with the weighting factors P

[t]Re(sk ) and P

[t]Im (sk ) given by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

P[t]Re(sk) =

[D[t−1](sk) ]∗

Re[D[t−1](sk) ]Re[N [t−1](sk) ] + Im[D[t−1](sk) ]Im[N [t−1](sk) ]

P[t]Im (sk) =

[D[t−1](sk)]∗

Re[D[t−1](sk)]Im[N [t−1](sk)]−Im[D[t−1](sk)]Re[N [t−1](sk)].

(8)It should be mentioned that in P

[t]Re(sk ) and P

[t]Im (sk ), a balanced

weighting is used to achieve comparable fitting precision forboth real parts and imaginary parts. The derivation of (8) isdetailed in Appendix. The algorithm of IRFA is given in Fig. 4,where the εiter is defined as the maximum error

εiter = maxk

{|Zfit(jωk ) − Zmeas(jωk )| / |Zmeas(jωk )|} .

(9)

The RFA method is first performed to provide a startingpoint, the IRFA is then applied until the stopping condition(s)(in this study: εmax = 10%, Nmax = 15) are fulfilled. To showthe efficiency of the proposed IRFA algorithm, different meth-ods are applied to fit the measured impedance (from 40 Hz to70 MHz) of an 8-turn planar inductor with Ferroxcube 3F3-Planar E38 ferrite core. A rational function like (1) with m =n = 6 is used for the fitting. It can be seen in Fig. 5(a) thatthe RFA fitting results are wrong on LF due to the frequencybias problem and the errors decrease when frequency increases.In Fig. 5(b) and (c), the IRFA methods alleviate the frequencybias problem in Fig. 5(a). As seen, better fitting accuracies areachieved at LF. The difference between Fig. 5(c) and (b) is the


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

implementation of balanced weighting. It can be seen from theerror curve shown in Fig. 5(c) that with balanced weighting, thefitting accuracy of the real part is further improved in compar-ison with Fig. 5(b). The precision of IRFA fitting is very goodon wide frequency band because the impedance curve is quitesmooth (very few resonances). However, the error may exceedthe specified error level εmax on the two extremities of fittingrange and the resonance frequencies. To achieve a better preci-sion, one can increase the order of the rational function at theexpense of obtaining a more bulky equivalent circuit. In order toobtain a simple equivalent circuit, the choice of the order turnsout to be a tradeoff between the accuracy and the complexity ofthe circuit.

C. Equivalent Circuit Synthesis

Based on the results obtained by the IRFA method, a system-atic approach is used for synthesizing the equivalent circuit [21].Representing (5) by the Foster expansion

Z(s) = d + e · s +Nr∑

i=1

ri

s − pi︸︷︷︸

real poles

+Nc∑

i=1

(ais + bi

s2 + mis + ni

)

︸︷︷︸complex conjugate pole pairs

.

(10)Note that the ai , bi , mi , and ni are different variables fromthose used in (1). As seen, the Z(s) can be expressed by thesum of a constant term d, an s-proportional term e, real pole-residue terms, and complex conjugate pole-residue pair terms.The same expression holds for the admittance Y (s). The con-stant d corresponds to a resistance and the e·s term correspondsto a capacitance for Y (s) or an inductance for Z(s), respectively.In Table I, the detailed equivalent circuit synthesis methods forreal-pole terms and complex-pole pair terms are summarized.

1) Real pole: with a positive residue ri , the equivalent cir-cuit is simply an R–L in series for Y (s) or R–C in parallel forZ(s), as shown in Table I. However, negative ri may appear in

the IRFA results, resulting in negative valued elements. Strictlyspeaking, negative elements are not feasible in real circuit de-sign so that they should be avoided. However, for EMI mod-eling and simulation purposes, these elements can be acceptedfor synthesizing an accurate equivalent circuit. Nonetheless, thenegative elements are difficult to handle during time-domainsimulations. To circumvent this difficulty, we suggest anothertopology: a positive-valued cell together with a negative resis-tance, as illustrated in Table I. Suppose that a negative residue ri

is extracted with a stable real pole pi (pi < 0) for the admittanceY (s). Normally, a negative R–L series cell is obtained. However,according to Table I, the circuit composed of a positive R–Cseries cell in parallel with a negative resistance −R can be used,where the expression for R, C, and −R are given by

⎧⎪⎨

⎪⎩

R = pi/ri positive

C = −1/(piR) positive

−R = −pi/ri negative.

(11)

This circuit is equivalent to the negative R–L series cell sincethey share the same rational function. The negative resistance−R are subsequently merged into the constant term d in (10),resulting in a new constant term d∗ = d − 1/R = d − ri/pi .Similar analysis can be done for the case of impedance Z(s).With these circuits, negative elements for real pole terms can bereduced to minimum.

2) Complex-pole pair: to treat complex-pole pair, a four-element circuit can be used [21]. It is the minimal type real-ization since there are four free variables ai , bi , mi , and ni inthe rational function (see Table I). Another equivalent circuitreferred as “extended” type, uses a six-element circuit but witheasier parameter calculation formulas [22]. In Fig. 6, the equiv-alent circuit corresponding to the admittance Y (s) in Fosterexpansion is illustrated.

As seen, the proposed IRFA together with the equivalentcircuit synthesis method enables generating a highly accurateHF model from a measured impedance (or admittance) with


TABLE IEQUIVALENT CIRCUIT SYNTHESIS FOR REAL POLES AND COMPLEX-POLE PAIRS

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

Fig. 7. Proposed HF model for CM chokes.

flexibility. Based the IRFA and the circuit synthesis method, theHF model for CM chokes will be introduced in the followingsection.

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. Itshould be noted that the parasitic capacitances are assumed to belossless. Besides, the capacitances Cs in Fig. 2(b) are discardedin our model since Ce , Cp , and Cc are sufficient for describingthe electrostatic behavior of a strongly coupled two-windingtransformer [13]. Instead of using a heuristic method to identify

TABLE IIMEASUREMENT CONFIGURATIONS FOR THE PROPOSED

EXTRACTION PROCEDURE

the leakage impedance Z1 and the magnetizing impedance Z2 ,the topologies of Z1 and Z2 of the proposed model are repre-sented by the equivalent circuit in Fig. 6, which are determinedby the results of the IRFA method. It is to be noted that thechoice of the topologies follows strict mathematical calculationso that the final equivalent circuit is guaranteed to be accurateover a wide frequency range.

B. Extraction Procedure

The extraction of the proposed HF model shown in Fig. 7begins with three specific impedance measurements T3 , T0 , andT2 , as shown in Table II. The IRFA method is then applied toextract the equivalent circuit for the impedances ZT 3 , ZT 0 , andZT 2 . Note that the measurement T1 in Fig. 3 is discarded sinceCs is removed from the model. Moreover, the measurement T4used in [15] is not necessary in the proposed procedure becauseT2 can provide enough information to extract the magnetizingimpedance Z2 . In order to validate the proposed method and the


Fig. 8. Measured impedances for the extraction. (a) ZT 3 . (b) ZT 0 . (c) ZT 2 .

extraction procedure, it is applied to model a planar CM chokewith eight turns on each winding. First, impedance measure-ments are carried out with an HP4294A precision impedanceanalyzer. The configurations T3 , T0 , and T2 are treated suc-cessively, as described in the following parts. Configuration T3represents a capacitive configuration which gives the value ofCc + Cp . The admittance YT 3 = 1/ZT 3 is fitted with the ex-pression

YT 3 = 1/ZT 3 = d + e · s (12)

where d denotes the dielectric losses and e denotes the capac-itance 2(Cc + Cp). The fitting starts from 100 Hz to guaranteea good measurement precision of capacitances. As the capaci-tance model is an R–C parallel cell, the fitting stops at 30 MHzto exclude the resonance, as shown in Fig. 8(a). The loss term dis neglected according to the lossless capacitance assumption.The extracted result for 2(Cc + Cp) is 29.98 pF.

Configuration T0 is a flux-subtracting configuration so onlythe leakage impedance Z1 /2 and the capacitance 2(Ce + Cp)are concerned. Again, to facilitate the circuit synthesis, the ad-mittance YT 0 = 1/ZT 0 is treated in the IRFA. In fact, the ca-pacitance 2(Ce + Cp) corresponds the e·s term in Fig. 6, so therational function YT 0(s) represented in the form of (1) mustfulfill: m − n = 1. In order to obtain a simple and accurateequivalent circuit, the values of m and n should be lowest pos-sible but still guarantee an acceptable accuracy. In this case, wechoose m = 6 and n = 5 after a few trials. In order to extract thedc resistance as well as the parasitic capacitance 2(Ce + Cp),the fitting frequency range covers from 40 Hz to 70 MHz, as

illustrated in Fig. 8(b). The fitting results for YT 0(s) are listedin Table III.

It can be seen from Table III that the extracted eT 0 is negative,resulting in a negative capacitance. The latter is not allowedbecause it will cause simulation instabilities [24]. In fact, thephase plot of ZT 0 [see Fig. 8(b)] shows that this impedance isnot capacitive beyond the resonance frequency fr ≈ 51 MHz.Around this frequency, the measured impedance ZT 0 is affectedby distributed effect because the winding length of the studiedCM choke is about 1 m, which is comparable to the wave lengthof the FR-4 based PCB structure at fr . Therefore, using thesimple parallel circuit shown in Table II to describe ZT 0 orYT 0 gives rise to the negative capacitance. To obtain a positivecapacitance, an approximation is performed. First, the complexpole-residue pair (pT 0 , rT 0) − (p∗T 0 , r

∗T 0), dT 0 = 3.0893E − 3

and eT 0 = −3.2803E − 12 terms of YT 0(s) in Table III arerecombined together to form a new admittance called YC T 0(s)

YC T 0(s) = dT 0 + eT 0 · s +rT 0

s − pT 0+

r∗T 0

s − p∗T 0. (13)

Indeed, the “true” parasitic capacitance is merged in YC T 0(s).The expression (13) is then simplified using Y ′

C T 0(s) = d′T 0 +e′T 0s by equating the values of YC T 0(s) and Y ′

C T 0(s) at theresonance frequency sr = j2πfr , as given by

dT 0 + eT 0 · sr +rT 0

sr − pT 0+

r∗T 0

sr − p∗T 0= d′T 0 + e′T 0 · sr .

(14)The obtained results are: e′T 0 = 3.22E − 11 and d′T 0 = 1.48E −03, which can be further represented by an R–C parallelcell. With this simplification, the positive parasitic capacitance2(Ce + Cp) and its parallel resistance are obtained. As the com-plex pole pair is thrown away during the approximation, thefitting precision of the real part of ZT 0 is impaired at highfrequencies, since the complex poles are indispensable for ob-taining high fitting precision around the resonance frequency,as shown in Fig. 9(a). However, tradeoffs have to be acceptedwith this lumped-element equivalent circuit.

The final circuit representation for impedance ZT 0 is shownin Fig. 10(b). The impedance Z1 /2 is contained in the dashedbox. The elements of the R–L cells are calculated with the realpole-residue pairs listed in Table III. These components can beinterpreted as the partial element equivalent circuits for model-ing the eddy current effect.

Configuration T2 is used as the last step of the procedureto identify the equivalent circuit of ZT 2 . A rough calculationis first performed at 100 kHz to estimate the CM inductancewith Im(ZT 2) = ωLCM/2, giving LCM = 382.4 μH. Similarto ZT 0 , the admittance form YT 2 is processed for the fitting,using a rational function with m = 6 and n = 5. As shownin Fig. 8(c), the fitting stops at 30 MHz, just before the secondresonance that cannot be modeled by the lumped-element equiv-alent circuit in Fig. 7. The fitting results are shown in Table III.According to the equivalent circuit T2 in Table II, eT 2 corre-sponds to the value of the parasitic capacitance 2(Ce + Cc)whereas the remaining parts (i.e., dT 2 , real poles and com-plex poles) represent the admittance 2(Z1 + Z2)−1 . With Z1


TABLE IIIFITTING RESULTS FOR THE PLANAR CM CHOKE

Fig. 9. Equivalent circuit extraction using IRFA. (a) ZT 0 . (b) ZT 2 .

Fig. 10. Extracted circuits for the CM choke. (a) ZT 3 . (b) ZT 0 . (c) ZT 2 .

being identified [see Fig. 10(b)], the magnetizing impedanceZ2 is calculated by subtracting the contribution of Z1 . TheIRFA is applied again to get the pole-residue information ofthe admittance Y2(s) = 1/Z2(s), as listed in Table III. Notethat a real pole p = −2.1616E + 08 with negative residuer = −9.4903E + 04 is extracted. This negative pole-residue istreated with the method presented in Table I, resulting in a

positive R(2.28 kΩ)–C(2.03 pF) series branch, which is inter-preted as the nonmagnetic HF losses in the ferrite material dueto the capacitive isolation of the grains [23]. The final equiva-lent circuit of Z2 /2 and ZT 2 is shown in Fig. 10(c). In Z2 , theL(190 μH)–R(4 mΩ) branch presents the dominant pole accord-ing the definition: a pole pi with its residue ri is called dominantpole if its Fi = |ri/Re(pi)| is much larger than the other poles.The values Fi for the poles of ZT 0 and Z2 are listed in Table IIIand the Fi of the L(190 μH)–R(4 mΩ) branch is much largerthan the others. Therefore, it has the most important influencein Z2 . The 190-μH inductance is in fact the CM filtering in-ductance because it is half the CM inductance LCM estimatedfrom ZT 2 . The simulated impedance of ZT 2 is compared withmeasured data in Fig. 9(b) in terms of real part and magnitude.As seen, the simulated results agree well with the measurementsup to 30 MHz.

With the parasitic capacitances obtained from the three con-figurations, three linear equations expressed in (15) can be estab-lished, giving the values of the stray capacitances: Cc = 6.1 pF,Ce = 7.2 pF, and Cp = 8.9 pF

⎧⎪⎨

⎪⎩

2Cc + 2Cp = 30 pF

2Ce + 2Cp = 32.2 pF

2Ce + 2Cc = 26.7 pF.

(15)

Till now the HF model for the studied planar CM choke is es-tablished. The obtained results will be validated by experimentaltests in next section.

V. EXPERIMENTAL VERIFICATION AND DISCUSSION

A. Experimental Verification

To validate the HF model and the procedure, the planar CMchoke is first examined by impedance measurements. Two mea-surement configurations are chosen: open circuit ZO and shortcircuit ZSC . These two configurations correspond to extremeloaded impedances for the component. Using SPICE, an acanalysis from 40 Hz to 30 MHz (Log Sweep) is performed onthe identified HF model and the simulation results are comparedwith measurements data, as shown in Fig. 11. It can be seen thatthe simulated curves match closely with the measured ones from40 Hz to 30 MHz.

As there are many small-valued elements in the model, thesensitivity of the model should be examined. Some qualitativeresults are given for this analysis.


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

1) The three parasitic capacitances Ce , Cp , and Cc affectthe resonance frequency and the response beyond theresonance.

2) In Z1 and Z2 , the branches of dominant poles determinethe behavior of the model on the inductive region. Themodel is less sensitive to the parameters in the braches ofnondominant poles.

3) The impedance around the resonance varies rapidly so theyare sensitive to the dominant poles branches, the complexpole-pairs branches and parasitic capacitances.

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

Next, an EMI filter using the topology shown in Fig. 1 isrealized with the planar CM choke, as shown in Fig. 12(a). Theleakage of the CM choke acts as DM inductors. The impedanceof the CM capacitors (4.7 pF) and the DM capacitor (68 nF) aremeasured by an impedance analyzer. The equivalent circuit forthe capacitors is given in Fig. 12(b) and their impedance ZC (s)can be expressed by

ZC (s) = RESR + sLESL +RP

sCRP + 1(16)

where RESR stands for equivalent series resistance and LESLfor equivalent series inductance. To get the values of the param-eters, (16) can be fitted by the IRFA method using a rationalfunction (1) with m = 2 and n = 1. The IL of the filter is mea-sured on an Agilent 5071C network analyzer using a four-portS parameter measurement method [9] [see Fig. 12(c) and (d)].

Fig. 12. IL measurements of the EMI filter. (a) EMI filter for test. (b) Equiv-alent circuit of capacitors. (c) Circuit of the measurement. (d) Measurementsetup.

The S parameter matrix [Sij , i, j = 1, 2, 3, 4] is measured from100 kHz to 30 MHz with logarithmic sweep. Postprocessing ofthis matrix is performed to obtain the mixed-mode S parameterScc21 and Sdd21 through the following equations:

⎧⎪⎨

⎪⎩

Scc21 =12

(S21 + S23 + S41 + S43)

Sdd21 =12

(S21 − S23 − S41 + S43)(17)

where Scc21 and Sdd21 give the CM and DM IL for the EMIfilter, respectively [9]. Moreover, a four-port S parameter simu-lation is performed over the same frequency range with the sameconfiguration as Fig. 12(c). The simulated CM IL Scc21 sim andDM IL Sdd21 sim are deduced by (17) as well. The comparisonbetween the measured ILs and the simulated ones are presentedin Fig. 13. It can be seen that the simulated result closely matchesthe measurement for CM. However, large difference at HF is ob-served for the DM. This is due to the parasitic coupling effectsamong the choke, capacitors, and trace loop [4], [11], which arenot considered in this model. However, this is out of the scopeof this paper.

B. Discussion

Through the IRFA method, accurate models for CM chokescan be built from impedance measurements. However, severalimportant aspects of the model and the procedure should beaddressed.

1) Passivity: as the whole extraction procedure treats passivemagnetic components, the passivity issues have to be con-sidered. The passivity of a component requires that thecomponent be dissipative in energy. However, the IRFAmethod cannot guarantee the passivity of the calculatedrational function. During the past ten years, many workshave been reported on the passivity verification and en-forcement techniques [24], [25]. In this paper, all the ex-tracted results are verified to be passive using the methodof [24]. In practice, as CM chokes are naturally dissipative


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

in HF due to core losses and copper losses, the passivityis usually respected if fitting precision is good enough.

2) Distributed effect: due to the distributed effect, manualadjustments are needed to avoid negative elements, whichrequire some experience about the equivalent circuit syn-thesis. This is the limit of using a lumped-element equiva-lent circuit model. The model presented in this paper is ef-fective up to 30 MHz, which corresponds to the frequencyrange of interest for the conducted emissions. Nonethe-less, it is necessary to consider the distributed effect tocover frequencies beyond 30 MHz, which requires imple-menting a distributed circuit model.

3) Saturation: CM chokes always work under flux bias whichwill cause saturation effect [26]. The proposed model isestablished on the basis of small signal measurements.Therefore, the error of the model will increase as the cur-rent increases. However, the primary focus of this paper isto propose a small signal HF model of CM chokes that canbe systematically synthesized. As the first step, the smallsignal model is now available and it can then be coupledwith nonlinear cell that accounts for the saturation of themagnetic material, which is the objective of a future study.

VI. CONCLUSION

In this paper, an HF equivalent circuit model of CM chokesused in the EMI filter is proposed incorporating a system-atic extraction procedure. The procedure uses an iterative ra-tional function approximation method to extract pole-residueform expressions from impedance measurements. Based on thepole-residue information, a systematic synthesis method is ap-plied to generate the equivalent circuit. The proposed proceduregives accurate models in a short time. This procedure is ap-

plied on a planar CM choke and is validated by simulations andmeasurements.

APPENDIX

According to the SK iteration, the denominator of the iterationt − 1D[t−1](sk ) is divided at both sides of the (3) of the iterationt, resulting in (6) and

N [t](sk )(D[t−1](sk ))∗

|D[t−1](sk )|2 =Zmeas(sk )D[t](sk )(D[t−1](sk ))∗

|D[t−1](sk )|2(18)

where the superscript ∗ denotes the complex conjugate. Formagnetic components, the imaginary part of the impedance(Z ≈ ωL) is much larger than its real part (losses) in the in-ductive region. This unbalance will lead to a low fitting qualityfor the real part during the least square process. In order toachieve a balanced fitting precision between the real part andthe imaginary part, the real and imaginary parts of (18) are nor-malized respectively by 1/Re[Z [t−1](sk )] and 1/Im[Z [t−1](sk )]as weighting factors

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1Re[Z [t−1](sk )]

=|D[t−1](sk )|2

Re[N [t−1](sk )(D[t−1](sk ))∗]

1Im[Z [t−1](sk )]

=|D[t−1](sk )|2

Im[N [t−1](sk )(D[t−1](sk ))∗].

(19)

Combining the SK iteration (18) with the weighting factors (19),the final formulations (7) and (8) are derived.

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Wenhua Tan was born in Jiangsu, China, in 1984.He received B.E. and M.E. degrees from the Xi’anJiaotong University, Xi’an, China, in 2006 and 2009,respectively, and the Eng. degree from the Ecole Cen-trale de Lyon, Ecully, France, in 2009. Since 2009, hehas been working toward the Ph.D. degree in electri-cal engineering in the Ecole Centrale de Lille, Univer-site Lille Nord de France, Villeneuve d’Ascq, France.

His main research interests include passive com-ponent integration, passive component modeling, andEMI filters design.

Carlos Cuellar (M’12) was born in Peru, in 1982.He received the Diploma degree in electronic engi-neering from the National University of San AntonioAbad, Cusco, Peru, in 2004, and the M.S. degree inelectrical engineering from the Universite des Sci-ences et Technologies de Lille, Villeneuve d’Ascq,France, in 2010, where he is currently working to-ward the Ph.D. degree.

In 2008, he was an Engineer in the MachupicchuHydroelectric plant, Peru. His current research in-terests include power conversion, passive EMI filter

design, and the characterization of magnetic materials used in power electronics.

Xavier Margueron (M’09) was born in Chambery,France, in 1980. He received the Diploma degreein electrical engineering from the Ecole NationaleSuperieure d’Ingenieurs Electriciens de Grenoble,Grenoble, France, in 2003, and the Doctorat de GenieElectrique from the Universite Joseph Fourier, Greno-ble, in 2006.

Since September 2007, he has been an AssociateProfessor at the Ecole Centrale de Lille, UniversiteLille Nord de France, Lille, France, where he carriesout research in the Laboratoire d’Electrotechnique et

d’Electronique de Puissance de Lille. His research interests include HF powerelectronics and passive component modeling.

Nadir Idir (M’93) received the Ph.D. degree electri-cal engineering from the University of Lille 1, France,in 1993.

He is currently a Full Professor with Instituts Uni-versitaires de Technologie A of the University of Lille1, where he teaches power electronics and electro-magnetic compatibility. Since 1993, he has been withthe Laboratory of Electrical Engineering and PowerElectronics, University of Lille 1. His research inter-ests include, design methodologies for HF switchingconverters, power devices (SiC and GaN), electro-

magnetic interference (EMI) in static converters, HF modeling of the passivecomponents, EMI filter design methodologies for switching converters.

Documents

A High Frequency Equivalent Circuit and Parameter Extraction Procedure for Common Mode Choke in the EMI Filter